ON FINITE GROUPS AND HOMOTOPY THEORY Ran Levi

1991 Mathematics Subject Classification. Primary 55R35; Secondary 55R40, 55Q52.

Author addresses:

Department of Mathematics, University of Rochester, Rochester, NY 14627 Current address: Mathemaisches Institut, Universität Heidelberg, Im Neuenheimer Feld

288, 69120 Heidelberg, Germany iv

Contents

Abstract vii

Preface ix

Acknowledgements xiii

Part 1 The Homology and Homotopy Theory Associated with ΩBπ_{p}^{∧}

Chapter 1. Introduction 3 1.1. Statement of Results 3 1.2. Organization of Part 1 7

Chapter 2. Preliminaries 9 2.1. Some Facts on the R-Completion Functor 9 2.2. Mod-R Acyclic Spaces and Proposition ?? 9 2.3. The Quillen “Plus” Construction 10

Chapter 3. A model for S_{∗}ΩX_{R}^{∧} 11 3.1. An Algebraic “Plus” Construction 11 3.2. Proof of Theorems ?? and ?? 14

Chapter 4. Homology Exponents for ΩBπ_{p}^{∧} 17 4.1. Extended Maps and Homotopies 17 4.2. Proof of Theorem ?? 19

Chapter 5. Examples for Homology Exponents 25 5.1. Groups with a Dihedral Sylow 2-Subgroup 25 5.2. Groups with a Semidihedral Sylow 2-Subgroup 27

Chapter 6. The Homotopy Groups of Bπ_{p}^{∧} 31 6.1. Some Basic Facts 31 6.2. Proof of Theorem ?? 33 6.3. Examples for Homotopy Exponents 34 Chapter 7. Stable Homotopy Exponents for ΩBπ_{p}^{∧} 37

7.1. Preliminaries on the Transfer 37 7.2. Proof of Theorem ?? 39 7.3. The Non-Existence of Exponents in π_{∗}^{s}Bπ_{p}^{∧} 42

Part 2 Finite Groups and Resolutions by Fibrations

Chapter 1. Introduction 49 1.1. Statement of Results 50 1.2. Organization of Part 2 52

Chapter 2. Preliminaries 53 2.1. Universal Central Extensions 53 2.2. Uniqueness of Homotopy Type, Special Case 53 2.3. Homotopy decomposition of Classifying Spaces 55 2.4. The Neisendorfer Fibre Square Lemma 56

Chapter 3. Resolutions by Fibrations 57 3.1. Definition and Basic Examples 57 3.2. A Fibration Lemma 59 3.3. The mod-p Cohen Conjecture 61

Chapter 4. Sporadic Examples 63 4.1. Groups with a Dihedral Sylow 2-Subgroup 63 4.2. Groups with a Semidihedral Sylow 2-Subgroup 73

Chapter 5. Groups of Lie Type and S-Resolutions 83 5.1. Preliminary Theorems 83 5.2. A Spherical Fibre Square 84 5.3. Proof of Theorem ?? 88 5.4. The Groups SL_{n}(𝔽_{q}) and Sp_{2n}(𝔽_{q}) 90 5.5. Proof of Theorem ?? and Examples 91

Chapter 6. Clark-Ewing Spaces and Groups 93 6.1. Construction 93 6.2. Spherical Resolutions of Loop Spaces on Clark-Ewing Spaces 96 6.3. Resolutions by Cohomological Considerations 96 6.4. Some Preliminaries from Representation Theory 99 6.5. Clark-Ewing Groups 99

Chapter 7. Discussion 103

References 105

Abstract

Let p be a fixed prime number. Let G denote a finite p-perfect group. We study the homotopy type of the p-completed classifying space BG_{p}^{∧}. The paper is divided into two parts. In part 1 we study the homology, homotopy and stable homotopy of ΩBG_{p}^{∧}, where G is a finite p-perfect group. We construct an algebraic analogue of the Quillen’s ”plus” construction for differential graded coalgebras. This construction is used to show that given a finite p-perfect group G, the loop spaces ΩBG_{p}^{∧} admits integral homology exponents. We give examples to show that in some cases our bound is best possible. We show that in general Bπ_{p}^{∧} admits infinitely many non-trivial k-invariants and discuss some examples where homotopy exponents exist. Finally classical constructions in stable homotopy theory are used to show that the stable homotopy groups of these loop spaces also have exponents.

In Part 2 we define the concept of resolutions by fibrations over an arbitrary family of spaces. We construct examples for finite p-perfect groups G such that ΩBG_{p}^{∧} is finitely resolvable over the family of spheres and their iterated loop spaces. In particular we get such resolutions for loop spaces associated with most finite groups of Lie type and with some of the polynomial cohomology spaces of Ewing and Clark. Various sporadic examples are also discussed.

Key words and phrases. Finite Groups, Completion and Localization, Loop Spaces, Exponents, Spherical resolvability.

The author was supported in part by NSF Grant #DMS-9013139.

viii ABSTRACT

Preface

The classifying spaces of discrete groups have been a subject for extensive study in the last two decades. We mention the remarkable result of H. Miller [48] giving a proof of the Sullivan conjecture. J. Lannes [36] studied mapping spaces out of a classifying space of an elementary abelian p-group and reproved Miller’s theorem. W. Dwyer and A. Zabrodsky [23, 62] studied maps between classifying spaces of compact Lie groups. S. Jackowsky and J. McClure [30] have obtained a homotopy decomposition for the p-completed classifying space of any compact Lie group. J. Martino and S. Priddy [43] refined the theory of [30] to obtain a criterion for p-local equivalence of classifying spaces of finite groups. E. Friedlander [27] has shown that there is a strong relationship between the classifying spaces of complex reductive Lie groups and the corresponding discrete groups of Lie type.

Regarding stable homotopy theory, G. Carlsson [13] has studied the stable homotopy theory of BG and obtained a proof of the Segal conjecture. P. May [47] later generalized Carlsson’s result. The Segal conjecture was also used by J. Martino and S. Priddy [40, 41, 42] in the study of stable homotopy theory associated with classifying spaces of finite groups. The list of authors who dealt with the various aspects of the homotopy theory associated with classifying spaces of finite groups and compact Lie groups is indeed too long to be spelled out.

In the early 1970’s D. Quillen introduced the “plus” construction and used it to define higher algebraic K- groups [54, 55]. The “plus” construction can be thought of as a method of associating with a space X and a perfect normal subgroup P of π_{1}(X), a new space X^{+} and a map i : X ⟶ X^{+} such that i induces a homology isomorphism with respect to any coefficients system on X^{+} and such that π_{1}(X^{+}) ≅ π_{1}(X)/P.

A natural question that arises at this point is whether one can go the other way, namely given a space X, does there exist a discrete group π such that H_{∗}(X) ≅ H_{∗}(Bπ) with respect to some coefficient system on X. Indeed, D. Kan and W. Thurston have established in [33] that given any connected space X with some reasonable properties, there exists a space T X together with a map t_{X} : T X ⟶ X inducing a homology isomorphism with respect to any coefficient system on X and such that T X is a space of type K(π, 1). Kan and Thurston also showed that the original homotopy type of X can be reconstructed from T X by means of the Quillen “plus” construction, i.e. T X^{+} ≃ X, where the “plus” construction is taken with respect to a particular perfect normal subgroup of π = π_{1}(T X). Unfortunately the Kan-Thurston construction produces a gigantic group as π_{1}(T X) even in the case where X is the simplest non-contractible space one can think of. The Kan-Thurston theorem does suggest however that understanding the “plus” construction on classifying spaces of various classes of groups might provide some interesting topological information.

In most applications, the Quillen “plus” construction on a space X is carried out with respect to a maximal normal perfect subgroup of π_{1}(X). In that case one might like to replace the (non-functorial) “plus” construction by the partial ℤ-completion functor of Bousfield and Kan or, in the case where π_{1}(X) is itself a perfect group, by the ℤ-completion functor [12].

For a commutative ring with unit R, let (−)_{R}^{∧} denote the R-completion functor of Bousfield and Kan. In particular, we write (−)_{p}^{∧}, (−)_{(p)} and (−)^{∧} for Rcompletion with R = 𝔽_{p}, ℤ_{(p)} and ℤ respectively. If X is a space such that H̃_{n}(X; ℤ) is finite for every n ≥ 0, then it is shown in [12] that X^{∧} ≃ ∏_{p} X_{p}^{∧}. Thus for such a space X, one can investigate the homotopy type of X^{∧} a prime at a time.

A particular class of spaces X, which has the property that H̃_{n}(X; ℤ) is a finite group for every n, is the class of classifying spaces of finite groups. Let B(−) denote the classifying space functor. Understanding the homotopy type of spaces of the form Bπ_{p}^{∧}, where π is a finite group is by and large the main theme of this paper.

Recall that a group π is said to be p-perfect if π is generated by its own commutator subgroup and p-th powers of elements of π. A group π is said to be perfect if it is p-perfect for every prime p dividing its order. Equivalently, we say that a group π is p-perfect if H_{1}(Bπ; 𝔽_{p}) = 0 and perfect if the same is true with integral coefficients.

Let π be a finite group and p, a fixed prime dividing the order of π. It is known that Bπ_{p}^{∧} is not in general the classifying space of a finite group. In fact, as we will show, spaces of this form tend to have rich higher homotopy groups.

It turns out that understanding the homotopy type of ℤ-completed classifying spaces of finite perfect groups, or more generally, the homotopy type of p-completed classifying spaces of finite p-perfect groups is almost equivalent to understanding the homotopy type of p-completed classifying spaces of all finite groups. We briefly explain this point.

Every finite group π fits in a short exact sequence as follows:

1 ⟶ O^{p}π ⟶ π ⟶ π/O^{p}π ⟶ 1

where O^{p}π is the minimal normal subgroup of π, such that π/O^{p}π is a p-group.

In particular, O^{p}π is the maximal normal p-perfect subgroup of π. Since an action of a p-group on another p-group is always nilpotent, it follows by [BK II.5] that the natural map

BO^{p}π_{p}^{∧} ⟶ Bπ_{p}^{∧}

is the universal covering space for Bπ_{p}^{∧}.

Thus in order to understand the p-completed classifying space of a finite group π, it makes sense to restrict attention the p-completion of BO^{p}π.

The starting point of our work is a paper of F. Cohen [17], in which it is shown that for many finite perfect groups π, the space ΩBπ^{∧} admits a certain finite resolution by fibrations over spheres and their loop spaces. Cohen had also observed [18] that some spaces of the form above tend to exhibit an interesting behavior in terms of other homotopy properties.

In this paper we discuss general results on spaces of the form Bπ_{p}^{∧} and their associated loop-spaces, where the group π is assumed p-perfect rather than perfect. We also extend the work of [17] by producing further examples of spaces of the form ΩBπ_{p}^{∧}, where π is a finite p-perfect group, which admit resolutions by fibrations over spheres and their loop spaces.

The paper is divided into two parts. In part 1 we discuss the classical algebraic invariants of spaces of the form ΩBπ_{p}^{∧} where π is a finite p-perfect group. We construct an analogue of the Quillen “plus” construction on the level of differential graded coalgebras and use it in order to show that for any finite p-perfect group π, the integral homology of ΩBπ_{p}^{∧} has an exponent given by the order of the Sylow p-subgroup of π. We also exhibit examples which show that the torsion in H_{∗}(ΩBπ_{p}^{∧}; ℤ_{(p)}) is sometimes larger than the torsion in H_{∗}(Bπ; ℤ_{(p)}).

The stable homotopy groups of spaces of the form ΩBπ_{p}^{∧}, where π is a finite p-perfect group, also present a similar behavior. We show that the order of the Sylow p-subgroup of π bounds the torsion in π_{∗}^{s}(ΩBπ_{p}^{∧}). We also consider examples where the torsion in the unstable homotopy groups of certain spaces of type ΩBπ_{p}^{∧}, for π finite and p-perfect, is bounded by the order of the Sylow psubgroup of π. On the other hand, we show that a space of the form ΩBπ_{p}^{∧}, where π is a finite p-perfect group, has infinitely many non-vanishing k-invariants. In contrast to the existence of stable homotopy exponents for spaces of the form ΩBπ_{p}^{∧}, we observe that for any finite group π, the stable homotopy of Bπ has elements of arbitrarily high order.

Part 2 is directly motivated by [17]. We recall the notion of resolvability by fibrations over an arbitrary family of topological spaces. In particular the family S of all spheres and their loop spaces is given special attention. We produce various examples of spaces of the form ΩBπ_{p}^{∧}, which are resolvable by fibrations over S.

Some curious results in part 2 appear in the study of ΩBπ_{2}^{∧} for certain finite groups π. We consider the homology of ΩBπ_{2}^{∧}, where π is a finite 2-perfect group and the Sylow 2-subgroup is either a dihedral or a semidihedral group.

The homology algebras in both cases turn out to be commutative and, in a sense that will be made precise, periodic.

Each part contains its own introduction and summary of results. We conclude this introduction by pointing out that our study provides some answers to questions one might be interested in, but even more open question, some of which will be spelled out, as we go along.

Acknowledgements

I take the pleasure to express my warm gratitude to the people who assisted me in accomplishing this work. Foremost I would like to thank my thesis advisor Fred Cohen for his careful guidance, unbelievable patience and enthusiasm and availability at all times.

I am also indebted to John Moore for many invaluable hours of private discussion. His willingness to listen to my ideas as well as to provide his outstanding mathematical knowledge and critical ear are greatly appreciated.

I extend my thanks to Joe Neisendorfer and John Harper for many interesting conversations and useful remarks. I would also like to thank Shiu Wong Dai Tamaki and Octav Cornea for always being there for a mathematical conversation.

xiv ACKNOWLEDGEMENTS

Part 1

The Homology and Homotopy Theory Associated with ΩBπ_{p}^{∧}

CHAPTER 1 Introduction

In this part we study spaces of type ΩBπ_{p}^{∧} in terms of some of their algebraic invariants. In particular we show that if π is a finite p-perfect group, then the integral homology and stable homotopy of ΩBπ_{p}^{∧} have exponents. The exponents that we find in both cases are given by the order of the Sylow p-subgroup of π.

1.1. Statement of Results

Theorem 1.1.1. Let π be a finite p-perfect group of order p^{r} · m, (m, p) = 1. Then

p^{r} · H̃_{∗}(ΩBπ_{p}^{∧}; ℤ_{(p)}) = 0.

It is known that if π is any finite group of order p^{r} · m, (p, m) = 1, then p^{r} annihilates the reduced p-primary homology of Bπ. However let X be an arbitrary 1-connected space whose p-primary homology has an exponent p^{r}. In general one would not expect p^{r} to annihilate the reduced p-primary homology of ΩX. In fact the only well known examples are spaces of the form ΩΣY, where ΣY is the reduced suspension of a connected space Y, with a p-primary homology exponent, and spaces of the form BS^{2n−1}{p^{r}}, where p and n are chosen such that S^{2n−1}{p^{r}} is an associative H-space. It is also not hard to observe that if X is a rationally contractible finite CW complex, then the reduced integral homology of ΩX has a finite exponent.

In particular if p is odd and n divides p − 1, then it is shown in [35], and more systematically in [17], that S^{2n−1}{p^{r}} ≃ ΩBπ_{p}^{∧}, where π is a certain finite p-perfect group whose Sylow p-subgroup is cyclic of order p^{r}. We also point out that spaces of type Bπ_{p}^{∧}, where π is a finite group whose order is divisible by p, are neither suspension spaces nor finite complexes, because of the existence of non-trivial products of arbitrary height in H^{∗}(Bπ_{p}^{∧}; 𝔽_{p}) [53]. Theorem 1.1.1 thus provides a large class of spaces with a rather special property. Furthermore, in chapter 5 below, we give examples where the Sylow 2-subgroup of π has order 2^{n}, H̃_{∗}(Bπ; ℤ_{(2)}) has exponent exactly 2^{n−1} but H̃_{∗}(ΩBπ_{2}^{∧}; ℤ_{(2)}) has exponent exactly 2^{n}.

Received by the editor November 10, 1993

Next we introduce the tools that were developed along the lines of proving theorem 1.1.1. In this paper a space always means a connected simplicial set or a CW complex. Let X be a space with a p-perfect fundamental group. The space X_{p}^{∧} can, in some sense, be thought of as a mod-p analogue of the Quillen “plus” construction on spaces with a perfect fundamental group. We realize this analogy on the level of differential graded coalgebras.

Proposition 1.1.1. Let C_{∗} be a connected differential graded coalgebra over a principal ideal domain R, which is dimensionwise projective over R. Let n be a positive integer. Then there exists a differential graded coalgebra, P_{n}C_{∗}, together with a map of differential graded coalgebras

i_{n} : C_{∗} ⟶ P_{n}C_{∗}

such that:

1. P_{n}C_{∗} is a quotient coalgebra of C_{∗} and i_{n} is the natural projection.

2. i_{n} is a split epimorphism in the category of differential graded R-modules.

3. i_{n} induces a homology isomorphism in dimensions k ≥ n + 1

4. P_{n}C is n-connected, i.e. (P_{n}C)_{i} = 0 for 1 ≤ i ≤ n. Furthermore, if C_{∗} itself is n-connected then i_{n} is the identity map.

For any simplicial set X and a ring R, let S_{∗}(X; R) denote the simplicial chains of X with coefficients in R. If X is a reduced simplicial set (i.e. a connected simplicial set with exactly one vertex) and R is a principal ideal domain, then S_{∗}(X; R) naturally satisfies the hypotheses of proposition 1.1.1.

Theorem 1.1.2. Let X be a reduced simplicial set. Let R be a principal ideal domain. Suppose that H_{1}(X; R) = 0. Then there exist quasi-isomorphisms of differential graded R-coalgebras

P_{1}S_{∗}(X; R) ⟶^{P_{1}(L)} P_{1}S_{∗}(X_{R}^{∧}; R) ⟵^{i} S_{∗}(X_{R}^{∧}; R).

Theorem 1.1.2 thus provides a model for S_{∗}(X_{R}^{∧}; R). We remark that in particular, the size of P_{1}S_{∗}(X; R) is significantly smaller than that of S_{∗}(X_{R}^{∧}; R). If X is a reduced Kan complex [46] then we have:

Theorem 1.1.3. Let X be a reduced Kan complex. Let R be a principal ideal domain. Suppose that H_{1}(X; R) = 0. Then there is an isomorphism of algebras over R:

H_{∗}(ΩX_{R}^{∧}; R) ≅ H_{∗}(\underline{Ω}P_{1}S_{∗}(X; R))

where \underline{Ω} denotes the Adams’ cobar construction.

Thus if X is a reduced Kan complex with a vanishing first mod-R homology then one may study the mod-R homology of ΩX_{R}^{∧} by looking at \underline{Ω}P_{1}S_{∗}(X; R).

Next we turn to study the homotopy groups of Bπ_{p}^{∧}. Here, a mod-p analogue of the acyclic functor of Dror [22], plays an important role. In particular, the theory of [22] and [12, VII.6] gives:

Proposition 1.1.2. Let R be a commutative ring with a unit. Let π be a group with H_{1}(Bπ; R) = 0. Then there is a principal π-bundle:

γ : L^{R}π ⟶ A^{R}Bπ (1.1.1)

such that 1. L^{R}π ≃ ΩBπ_{R}^{∧}

2. A^{R}Bπ is R-acyclic, i.e. H̃_{∗}(A^{R}π; R) = 0.

3. π acts trivially on the homotopy groups of L^{R}π.

If π is a p-perfect group and R = ℤ_{(p)}, then the hypotheses of proposition 1.1.2 are satisfied. We use this idea to gain information on the homotopy type of ΩBπ_{p}^{∧}.

Theorem 1.1.4. Let π be a finite p-perfect group. Then the Postnikov tower for ΩBπ_{p}^{∧} has infinitely many non vanishing k-invariants. In particular π_{∗}Bπ_{p}^{∧} is non-trivial in arbitrarily high dimensions.

Various facts that were discovered, while this work was in progress, raised the conjecture that if π is finite and p-perfect then π_{∗}Bπ_{p}^{∧} has an exponent, possibly bounded above by the order of the Sylow p-subgroup of π (see section 6.3). A stable version of this statement turns out to be true. The proof of the following theorem is based upon the observation that the stable transfer map for the bundle (1.1.1) in proposition 1.1.2 is null homotopic.

Theorem 1.1.5. Let π be a finite p-perfect group of order p^{r} · m, (p, m) = 1. Then

p^{r} · π_{∗}^{s}ΩBπ_{p}^{∧} = 0.

Theorem 1.1.5 might be surprising in view of the following fact, which deals with the stable homotopy of Bπ_{p}^{∧} rather than that of ΩBπ_{p}^{∧}. Proposition 1.1.3 below is likely to be well known to the experts, but stands in contrast to theorem 1.1.5.

Proposition 1.1.3. Let π be a finite group and p a prime dividing the order of π. Then π_{∗}^{s}Bπ_{p}^{∧} has p-torsion of arbitrarily high order.

1.2. Organization of Part 1

This part of the paper is organized as follows: In chapter 2 we state some preliminaries on the R-completion functor of Bousfield and Kan. We also discuss mod-R acyclic spaces and deduce proposition 1.1.2. Finally we relate the Zcompletion functor to the Quillen “plus” construction. The material included in chapter 2 is essential for the rest of the paper. However the reader who is familiar with [12] may safely skip it. In chapter 3 we prove proposition 1.1.1, and theorems 1.1.2 and 1.1.3. In chapter 4, we use theorem 1.1.3 to prove theorem 1.1.1. chapter 5 contains a few examples where the actual homology exponent of ΩBπ_{p}^{∧} is computed. In chapter 6 we prove theorem 1.1.4 and give some explicit examples for the existence of unstable homotopy exponents. In chapter 7 we prove theorem 1.1.5 and proposition 1.1.3.

8 1. INTRODUCTION

CHAPTER 2 Preliminaries

Unless otherwise specified, a space X will mean a pointed, connected simplicial set or a CW-complex. By a ring R, we mean a commutative ring with unit.

2.1. Some Facts on the R-Completion Functor

Let (−)_{R}^{∧} denote the R-completion functor of Bousfield and Kan [12]. For R = 𝔽_{p} or ℤ_{(p)}, we denote (−)_{R}^{∧} by (−)_{p}^{∧} or (−)_{(p)} respectively.

Recall from [12] that a space X is said to be R-good if the R-completion map X ⟶ X_{R}^{∧}, induces an isomorphism in R-homology. If X is a space whose reduced integral homology groups are all finite, then X is R-good for R = 𝔽_{p} or R ⊆ ℚ. Thus for such a space X, one has [12, VII,4.2]

X^{∧} ≃ ∏_{p} X_{p}^{∧}

A space X is said to be R-perfect, if H_{1}(X; R) = 0. In particular, we say that X is p-perfect if it is R-perfect for R = 𝔽_{p} or ℤ_{(p)}. A group π is said to be R-perfect if Bπ is an R-perfect space. R-perfect spaces are also R-good, for R = 𝔽_{p} or R ⊆ ℚ. Let p be a prime. A space X is said to have finite p-type if H_{n}(X; ℤ_{(p)}) is a finite abelian p-group for any n ≥ 1. Let X be a p-perfect space of finite p-type. The p-completion map, L_{p} : X ⟶ X_{p}^{∧} induces an 𝔽_{p}-homology isomorphism. Moreover, because of finite p-type, this is equivalent to L_{p} inducing a ℤ_{(p)}-homology isomorphism. Since X_{p}^{∧} is p-local we get equivalences:

X_{(p)} ≃ (X_{p}^{∧})_{(p)} ≃ X_{p}^{∧}.

Notice that classifying spaces of finite p-perfect groups are examples for p-perfect spaces of finite p-type.

2.2. Mod-R Acyclic Spaces and Proposition 1.1.2

Theorem 2.2.1 (Bousfield, Kan). Let R be a commutative ring with a unit. Let X be a pointed connected R-good space. Let

α : A^{R}X ⟶ X

be the fibration induced by ϕ : X ⟶ X_{R}^{∧} from the path-loop fibration over X_{R}^{∧}. Then A^{R}(−) is a functor with the properties:

1. H̃_{∗}(A^{R}X; R) = 0 and

2. The map α : A^{R}X ⟶ X is a weak equivalence if and only if H̃_{∗}(X; R) = 0

From theorem 2.2.1 it follows that α : A^{R}X ⟶ X is universal among maps of R-acyclic spaces into X. For R = ℤ, A^{R}(−) coincides with the acyclic functor of Dror [22]. Dror’s functor is given as the inverse limit of a tower of principal fibrations over X. In fact if R ⊆ ℚ, then it is possible to use Dror’s method to construct a mod-R acyclic functor that, up to homotopy, coincides with A^{R}(−) of [12]. For R = ℤ_{(p)}, we denote the functor A^{R}(−) by A^{p}(−).

We are now ready to prove proposition 1.1.2. Let π be an R-perfect group. By theorem 2.2.1, there is a principal fibration:

ΩBπ_{R}^{∧} ⟶ A^{R}Bπ ⟶^{α} Bπ

In particular π = π_{1}Bπ acts trivially on π_{∗}ΩBπ_{R}^{∧}.

Let γ : L^{R}π ⟶ A^{R}Bπ be the principal π bundle induced by α from the universal bundle over Bπ. Obviously L^{R}π ≃ ΩBπ_{R}^{∧}. This proves 1 and 2.

For 3, observe that the action of π on L^{R}π by deck transformations coincides, up to homotopy, with the action of π_{1}Bπ on ΩBπ_{R}^{∧}. This completes the proof.

2.3. The Quillen “Plus” Construction

Let X be a connected space. Let P ◁ π_{1}X be a perfect normal subgroup. The Quillen “plus” construction with respect to P associates with X a new space X^{+}, together with a map

i : X ⟶ X^{+}

such that

1. i induces a homology isomorphism (with respect to any coefficient system for X^{+}) and

2. π_{1}X^{+} ≅ π_{1}X/P.

The “plus” construction on X is obtained from X by attaching cells [2, 6]. Thus, in particular, it is not functorial, although one can show that the resulting space X^{+} is unique, up to homotopy. If X is Z-perfect, then the universal property of the Bousfield-Kan Z-completion functor (−)^{∧} implies that X^{∧} ≃ X^{+}.

CHAPTER 3 A model for S_{∗}ΩX_{R}^{∧}

In this chapter we introduce a model for the chain algebra S_{∗}ΩX_{R}^{∧}, the tool which was developed in order to prove theorem 1.1.1. It seems likely that our model should have further applications.

Throughout this chapter, R is a fixed principal ideal domain. By a DGRcoalgebra we shall always mean a connected differential graded coalgebra over R, which is dimensionwise projective over R. A DGR-coalgebra, C_{∗}, is said to be perfect if H_{1}(C_{∗}) = 0. Through the end of this chapter a space will mean a simplicial set, although much of the material presented here does not depend on this requirement.

The idea for the construction described here comes from the equivalence, discussed in the previous chapter, X^{+} ≃ X^{∧}, for any space X with a perfect fundamental group. The motivating example is:

Example 3.0.1. Let π be an R-perfect group. Let Bπ be the classifying space of π and B[π] be the classifying construction for the group ring R[π]. Then Bπ is a R-perfect space and B[π] has a natural structure of a perfect DGR-coalgebra.

3.1. An Algebraic “Plus” Construction

Here we mimic the Quillen “plus” construction on an algebraic level. We restate proposition 1.1.1 as:

Proposition 3.1.1. Let C_{∗} be a DGR-coalgebra. Then for every n ≥ 1, there exists a DGR-coalgebra, P_{n}C_{∗}, together with a map of DGR-coalgebras

i_{n} : C_{∗} ⟶ P_{n}C_{∗}

such that:

1. P_{n}C_{∗} is a quotient coalgebra of C_{∗} and i_{n} is the natural projection.

2. i_{n} is a split epimorphism in the category of differential graded R-modules.

12 3. A MODEL FOR S_{∗}ΩX_{R}^{∧}

3. i_{n} induces a homology isomorphism in dimensions k ≥ n + 1

4. P_{n}C is n-connected, i.e. (P_{n}C)_{i} = 0 for 1 ≤ i ≤ n. Furthermore, if C_{∗} itself is n-connected then i_{n} is the identity map.

Proof. Since R is a principal ideal domain, the following is a short exact sequence of projective R-modules

0 ⟶ Z_{n+1}(C_{∗}) ↪ C_{n+1} ⟶^{d} B_{n}(C_{∗}) ⟶ 0.

Since B_{n}(C_{∗}) is projective, the differential d, has a right inverse

σ : B_{n}(C_{∗}) ⟶ C_{n+1}.

Let C̃_{n+1} denote the image of σ in C_{n+1}. The sequence

⋯ ⟶ 0 ⟶ C̃_{n+1} ⟶ C_{n} ⟶ ⋯ ⟶ C_{1} ⟶ 0

is easily seen to be a differential coideal in C_{∗}. Denote it by I_{∗}^{n}. Define P_{n}C_{∗} = C_{∗}/I_{∗}^{n}. Let i_{n} be the natural projection. Notice that if C_{∗} is n-connected then I_{∗}^{n} = 0, so i_{n} is the identity map.

Clearly P_{n}C_{∗} is n-connected. Furthermore, as a chain complex, I_{∗}^{n} is exact in dimensions k ≥ n + 1. Using the long exact homology sequence for the short exact sequence of chain complexes:

0 ⟶ I_{∗}^{n} ⟶ C_{∗} ⟶ P_{n}C_{∗} ⟶ 0

one immediately verifies 3.

Finally, define η : P_{n}C_{∗} ⟶ C_{∗} by

η(b̄_{m}) = \{ \begin{matrix}0 & if m ≤ n \\ b_{m} & if m ≥ n + 1 or m = 0 \\ b_{m} − σd(b_{m}) & if m = n + 1 \end{matrix}

where b̄_{m} ∈ (P_{n}C_{∗})_{m} is the equivalence class of b_{m} ∈ C_{m}. One routinely checks that η is a chain map and that i_{n} ∘ η = 1_{P_{n}C_{∗}}. This completes the proof.

We point out that i_{n} above, is not a split epimorphism in the category of DGR-coalgebras, but only in the category of DGR-modules. When no confusion can arise we shall denote i_{n} simply by i.

Corollary 3.1.1. Let C_{∗} be a DGR-coalgebra with H_{j}C_{∗} = 0 for 1 ≤ j ≤ n. Then i_{∗} : H_{∗}C_{∗} ⟶ H_{∗}P_{k}C_{∗} is an isomorphism for every k ≤ n.

The construction P_{n}(−) is not functorial, since its definition involves a choice. However one does have:

Proposition 3.1.2. Let f : C_{∗} ⟶ D_{∗} be a map of DGR-coalgebras. Let n be a positive integer. Suppose that B_{n}(f) : B_{n}(C_{∗}) ⟶ B_{n}(D_{∗}) is a split monomorphism. Then there exists a choice of right inverses, σ : B_{n}(C_{∗}) ⟶ C_{n+1} and σ′ : B_{n}(D_{∗}) ⟶ D_{n+1} for the respective differentials, such that f induces a map of DGR-coalgebras, P_{n}(f) : P_{n}C_{∗} ⟶ P_{n}D_{∗} between the respective P_{n}(−) constructions. Furthermore, the following diagram commutes:

[figure: C_{∗}; i_{n}; D_{∗}; i_{n}; f; P_{n}C_{∗}; P_{n}D_{∗}; P_{n}(f)]

Proof. We need only to show that, under our assumptions, one can choose σ and σ′ such that

[figure: C_{n+1}; f; B_{n}(C_{∗}); f; σ; D_{n+1}; B_{n}(D_{∗}); σ′]

commutes. This readily follows from lemma 3.1.1 below.

Define a category PR as follows: Objects are triples (A, f, B), where A and B are R-modules and f : A ⟶ B is a homomorphism. Morphisms in PR are commutative squares.

Lemma 3.1.1. Let R be a principal ideal domain. Then objects in PR, of the form (P, j, Q), where P and Q are projective over R and j a split monomorphism, are projective in PR.

Proof. Let (P, j, Q) ∈ PR be an object satisfying the conditions of the lemma. Then the short exact sequence:

0 ⟶ P ⟶^{j} Q ⟶^{p} Coker(j) ⟶ 0

is split. Let i be a left inverse for j and q be a right inverse for p.

Let (α, β) : (A, f, B) ⟶ (P, j, Q) be an epimorphism in PR. Then α and β are epimorphisms of R-modules. Define (γ, δ) : (P, j, Q) ⟶ (A, f, B) as follows: Let γ be an arbitrary right inverse for α. Let δ′ be an arbitrary right inverse for β. Define δ : Q ⟶ B by the formula:

δ(x) = (f ∘ γ ∘ i + δ′ ∘ q ∘ p)(x)

One routinely verifies that (γ, δ) is a right inverse for (α, β). The result follows.

Example 3.1.1. Let A and X be reduced spaces (i.e. simplicial sets with only one vertex). Let f : A ⟶ X be a monomorphism. Then S_{∗}(X; R) has a natural structure of a DGR-coalgebra, and the induced map, f_{∗} : S_{∗}(A; R) ↪ S_{∗}(X; R), maps S_{∗}(A; R) injectively into a sub-DGR-coalgebra. The map f_{n} : S_{n}(A; R) ↪ S_{n}(X; R) is a split monomorphism for every n. Suppose further that H_{1}X = H_{1}A = 0. Then one also has S_{1}(X; R) = B_{1}(S_{∗}(X; R)) and S_{1}(A; R) = B_{1}(S_{∗}(A; R)). Thus the assumptions of proposition 3.1.2 are satisfied.

Finally we remark that although P_{n}C_{∗} depends upon the choice of the right inverse σ, its homotopy type as a chain complex is unique, whenever i_{n} induces an isomorphism in homology. Indeed, in that case, the mapping cone for i_{n} is acyclic. Since both C_{∗} and P_{n}C_{∗} are dimensionwise projective over R, so is the mapping cone. Hence the mapping cone is contractible, which is equivalent to i_{n} being a chain equivalence.

3.2. Proof of Theorems 1.1.2 and 1.1.3

Let X be a reduced space. Then S_{∗}(X; R) has a natural structure of a DGRcoalgebra. Furthermore, the natural map S_{∗}(X; R) ⟶ S_{∗}(X_{R}^{∧}; R) is a map of DGR-coalgebras. The theory of section 3.1 now yields theorem 1.1.2.

By [12, I.5.5], the R-completion map L : X ⟶ X_{R}^{∧} is a monomorphism. Furthermore both X and X_{R}^{∧} are reduced R-perfect spaces. Thus X and X_{R}^{∧} fit in the context of example 3.1.1. Hence by proposition 3.1.2, there is a map of DGR-coalgebras

P_{1}(L) : P_{1}S_{∗}(X; R) ⟶ P_{1}S_{∗}(X_{R}^{∧}; R)

which obviously induces a homology isomorphism. Since both P_{1}S_{∗}(X; R) and P_{1}S_{∗}(X_{R}^{∧}; R) are dimensionwise projective over R this is equivalent to P_{1}(L) being a chain equivalence. Since X_{R}^{∧} is simply-connected, it follows that the projection

i_{1} : S_{∗}(X_{R}^{∧}; R) ⟶ P_{1}S_{∗}(X_{R}^{∧}; R)

is also a chain equivalence, This completes the proof of theorem 1.1.2.

For theorem 1.1.3 we assume in addition that X is a Kan complex [46]. Thus we may consider the Eilenberg subcomplexes of X [24].

Let E_{1}X_{R}^{∧} denote the first Eilenberg subcomplex of X_{R}^{∧}. The composite

S_{∗}(E_{1}X_{R}^{∧}; R) ↪ S_{∗}(X_{R}^{∧}; R) ⟶^{i_{1}} P_{1}S_{∗}(X_{R}^{∧}; R)

3.2. PROOF OF THEOREMS 1.1.2 AND 1.1.3 15

consists of maps of DGR-coalgebras and induces an isomorphism in homology. Thus it is a quasi isomorphism of DGR-coalgebras. Clearly the same is true for

P_{1}(L) : P_{1}S_{∗}(X; R) ⟶ P_{1}S_{∗}(X_{R}^{∧}; R).

Thus, the induced maps

\underline{Ω}S_{∗}(E_{1}X_{R}^{∧}; R) ⟶ \underline{Ω}P_{1}S_{∗}(X_{R}^{∧}; R) ⟵ \underline{Ω}P_{1}S_{∗}(X; R)

are quasi isomorphisms of DGR-algebras. But by [1]

H_{∗}(\underline{Ω}S_{∗}(E_{1}X_{R}^{∧}; R)) ≅ H_{∗}(ΩX_{R}^{∧}; R)

as algebras over R. This completes the proof of theorem 1.1.3.

16 3. A MODEL FOR S_{∗}ΩX_{R}^{∧}

CHAPTER 4 Homology Exponents for ΩBπ_{p}^{∧}

In this chapter we apply the machinery developed above to prove theorem 1.1.1. Let π be a finite p-perfect group and Bπ be the classifying space for π. Throughout this chapter, let R denote a fixed ℤ-torsion free, p-local principal ideal domain (e.g. ℤ_{(p)}, ℤ_{p}^{∧}). The space Bπ can naturally be considered as a reduced Kan complex. The simplicial chain complex S_{∗}(Bπ; R) is isomorphic to the reduced classifying construction on the group ring R[π], as a DGR-coalgebra. By abuse of notation we omit the R and denote the classifying construction simply by B[π]. Since π is p-perfect and R, p-local, it follows that B[π] is a perfect DGR-coalgebra.

By theorem 1.1.3, there is an isomorphism of algebras:

H_{∗}(\underline{Ω}P_{1}B[π]) ≅ H_{∗}(ΩBπ_{R}^{∧}; R)

For any R-module, M, and an element r ∈ R, let ϕ_{r} : M ⟶ M be the map given by multiplication by r. Theorem 1.1.1 is proven by showing that

ϕ_{|π|} : \underline{Ω}P_{1}B[π] ⟶ \underline{Ω}P_{1}B[π]

is null-homotopic.

4.1. Extended Maps and Homotopies

Throughout this section by DGR-module we mean a differential graded Rmodule. Similarly, by a DGR-algebra we mean a connected differential graded algebra over R, which is dimensionwise free over R.

Let M be a connected graded R-module which is dimensionwise free over R. Let T (M) denote the tensor algebra on M. Then T (M) is also dimensionwise free over R. A choice of a set of free generators for M over R, gives rise to a canonical set of free generators for T (M) as an R-module.

Following the terminology of J. C. Moore, recall that a map over a category of graded objects is a map of some degree between objects in the category. In particular maps in the category are exactly those maps of degree zero.

18 4. HOMOLOGY EXPONENTS FOR ΩBπ_{p}^{∧}

Definition 4.1.1. Let T be a graded R-algebra. Let s : T ⟶ T be a map over the category of graded R-modules. We say that s is right extended if s is also a map of right T -modules, i.e. s(xy) = s(x)y for all x, y ∈ T.

Lemma 4.1.1. Let M be a graded R-module which is dimensionwise free over R. Let j : M ⟶ T (M) be the natural inclusion into tensors of length one. Let s′ : M ⟶ T (M) be a map over the category of graded R-modules. Then there exists a unique map s : T (M) ⟶ T (M) such that s is right extended and s ∘ j = s′.

Proof. For any x ∈ M let [x] denote j(x) ∈ T (M). Let [x_{1} | x_{2} | ⋯ | x_{n}] denote the product [x_{1}] · [x_{2}] ⋯ [x_{n}] in T (M). Let {y_{i}}_{i∈I} be a set of free generators for M over R. Define

s[y_{1} | y_{2} | ⋯ | y_{n}] = (s′y_{1}) · [y_{2} | ⋯ | y_{n}]

and extend by linearity.

Since both M and T (M) are dimensionwise free over R, it follows by construction, that s is right extended. For every x ∈ M, we have:

s ∘ j(x) = s[x] = s′(x).

Finally, uniqueness follows from the fact that Im(j) generates T (M) freely as an algebra. Thus any map s_{1} which coincides with s on Im(j) and is in addition a map of right T (M)-modules must be identical to s.

Let B_{∗} be a simply connected DGR-coalgebra. Let J(B_{∗}) denote the kernel of the augmentation map. The cobar construction \underline{Ω}B_{∗}, as an algebra, is the tensor algebra T (Σ^{−1}J(B_{∗})). As a chain complex \underline{Ω}B_{∗} is a bicomplex with differentials D_{I} and D_{E}, defined on generators by:

1. D_{I}[x] = −[d(x)], where d is the differential of B_{∗}.

2. D_{E}[x] = ∑_{i}(−1)^{|x′_{i}|}[x′_{i}|x′′_{i}], where the reduced diagonal map on x is given by Δ̄x = ∑_{i} x′_{i} ⊗ x′′_{i}.

The differentials D_{I} and D_{E} are defined on arbitrary elements by the requirement that they are derivations of the algebra structure. The total differential D is given by D = D_{I} + D_{E} [1].

Lemma 4.1.2. Let B_{∗} be a simply connected DGR-coalgebra. Let s : \underline{Ω}B_{∗} ⟶ \underline{Ω}B_{∗} be a right extended map over the category of graded R-modules of odd degree n > 0. Let φ : \underline{Ω}B_{∗} ⟶ \underline{Ω}B_{∗} be a right extended map over the category of DGR-modules of degree n − 1. Then s is a null-homotopy for φ if and only if (ds + sd)[x] = φ[x], ∀xεΣ^{−1}J(B_{∗}).

Proof. The “only if” direction is obvious. For the “if” direction it suffices to show that under our assumption sd + ds = φ on generators [x_{1} | ⋯ | x_{n}]. Indeed

(ds + sd)[x_{1} | ⋯ | x_{n}] =
(ds + sd)([x_{1}] · [x_{2} | ⋯ | x_{n}]) =
(ds[x_{1}]) · [x_{2} | ⋯ | x_{n}] + (−1)^{|[x_{1}]|+n} · s[x_{1}] · d[x_{2} | ⋯ | x_{n}] +
(sd[x_{1}]) · [x_{2} | ⋯ | x_{n}] + (−1)^{|[x_{1}]|} · s[x_{1}] · d[x_{2} | ⋯ | x_{n}] =
((ds + sd)[x_{1}]) · [x_{2} | ⋯ | x_{n}] =
(φ[x_{1}]) · [x_{2} | ⋯ | x_{n}] =
φ[x_{1} | ⋯ | x_{n}]

and the proof is complete.

4.2. Proof of Theorem 1.1.1

The crucial step in the proof is the construction of a null-homotopy for ϕ_{|π|}. Let B[π] be, as before, the reduced classifying construction for a finite p-perfect group π over a ℤ-torsion free p-local principal ideal domain R. Let E[π] denote the reduced bar resolution of the trivial π-module R. Thus E[π]⊗_{π}R≅ B[π]. The complex E[π] is a contractible chain complex with a canonical contraction c : E[π] ⟶ E[π], given by

c(g[g_{1} | ⋯ | g_{n}]) = [g | g_{1} | ⋯ | g_{n}].

Let q : E[π] ⟶ B[π] be the natural projection. Let tr : B[π] ⟶ E[π] be the transfer map given by

tr[g_{1} | ⋯ | g_{n}] = ∑_{gεπ} g[g_{1} | ⋯ | g_{n}].

Then the map s : B[π] ⟶ B[π], given by s = q ∘ c ∘ tr is easily checked to be a null homotopy of ϕ_{|π|} : B[π] ⟶ B[π].

Let σ : B_{1}[π] ⟶ B_{2}[π] be a right inverse of the differential. Construct P_{1}B[π] with respect to σ. Let i : B[π] ⟶ P_{1}B[π] be the projection. Recall (Proposition 1.1.1) that i is a split epimorphism in the category of differential graded R-modules. Let η : P_{1}B[π] ⟶ B[π] be a right inverse of i. Then the map s̃: P_{1}B[π] ⟶ P_{1}B[π], given by, s̃ = i ∘ s ∘ η is a null-homotopy of ϕ_{|π|} on P_{1}B[π].

Let ŝ denote the composite

Σ^{−1}J(P_{1}B[π]) ⟶^{s̃} Σ^{−1}J(P_{1}B[π]) ↪ \underline{Ω}P_{1}B[π].

Let S_{I} : \underline{Ω}P_{1}B[π] ⟶ \underline{Ω}P_{1}B[π] denote the unique extension of −ŝ to \underline{Ω}P_{1}B[π] in the sense of lemma 4.1.1.

Define ε : Σ^{−1}J(P_{1}B[π]) ⟶ \underline{Ω}P_{1}B[π] as follows:

ε(x) = 0 for | x |= 1 and
ε[g_{1} | ⋯ | g_{n}] = [i s_{2} σ[g_{1}]] · [i([g_{2} | ⋯ | g_{n}])] for n > 2.

where s_{2} is the second component of s : B[π] ⟶ B[π], defined above. Let S_{E} : \underline{Ω}P_{1}B[π] ⟶ \underline{Ω}P_{1}B[π] denote the unique extension of ε to \underline{Ω}P_{1}B[π], in the sense of lemma 4.1.1.

Define S : \underline{Ω}P_{1}B[π] ⟶ \underline{Ω}P_{1}B[π] by:

S = S_{I} + S_{E}.

Theorem 1.1.1 follows by showing that S defined above is a null-homotopy for ϕ_{|π|}. In fact we have:

Proposition 4.2.1. Let π be a finite p-perfect group. Let D_{I}, D_{E}, S_{I} and S_{E} be as above. Then for every element x ∈ Σ^{−1}J(P_{1}B[π]), the following relations hold:

1. (D_{I}S_{I} + S_{I}D_{I})[x] = ϕ_{|π|}[x]

2. (D_{E}S_{E} + S_{E}D_{E})[x] = 0

3. (D_{E}S_{I} + S_{I}D_{E})[x] = −(D_{I}S_{E} + S_{E}D_{I})[x]

Proof. We collect, from above, the definitions of some maps, which will take part in the computation:

Recall that i : B[π] ⟶ P_{1}B[π] is a split epimorphism in the category of DGR-modules with a right inverse η. For i(x) ∈ (P_{1}B[π])_{2}, we have η(i(x)) = x − σd(x) , where σ : B_{1}[π] ⟶ B_{2}[π] is a right inverse for the differential (see proposition 3.1.1). The map s : B[π] ⟶ B[π] is the canonical null-homotopy of ϕ_{|π|}. The “induced” map s̃ : P_{1}B[π] ⟶ P_{1}B[π] is given as the composite s̃ = i ∘ s ∘ η. With this information in mind, we are ready to start the proof.

1. Let x ∈ Σ^{−1}JP_{1}B[π]. Then

(D_{I}S_{I} + S_{I}D_{I})[x] =
−D_{I}[s̃(x)] − S_{I}[d(x)] =
[ds̃(x)] + [s̃d(x)] =
[(ds̃ + s̃d)(x)] =
[ϕ_{|π|}(x)] =
ϕ_{|π|}[x]

2. Let x = i([g_{1}| ⋯ |g_{n}]) ∈ Σ^{−1}JP_{1}B[π], g_{i} ∈ π and n > 2. Then

D_{E}S_{E}[x] =

D_{E}([is_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])]) = [is_{2}σ([g_{1}])] · D_{E}[i([g_{2}|g_{3}| ⋯ |g_{n}])]

On the other hand:

S_{E}D_{E}[x] = S_{E}D_{E}[i([g_{1}|g_{2}| ⋯ |g_{n}])] =

S_{E} (∑_{k=2}^{n−2}(−1)^{k}[i([g_{1}|g_{2}| ⋯ |g_{k}])] · [i([g_{k+1}|g_{k+2}| ⋯ |g_{n}])]) =

∑_{k=3}^{n−2}(−1)^{k}[is_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{k}])] · [i([g_{k+1}|g_{k+2}| ⋯ |g_{n}])] =

[is_{2}σ([g_{1}])] · ∑_{k=3}^{n−2}(−1)^{k}[i([g_{2}|g_{3}| ⋯ |g_{k}])] · [i([g_{k+1}|g_{k+2}| ⋯ |g_{n}])] =

−[is_{2}σ([g_{1}])] · D_{E}[i([g_{2}|g_{3}| ⋯ |g_{n}])] = −D_{E}S_{E}[x]

Observe that if n = 2, then D_{E}S_{E}[[g_{1}|g_{2}]] = S_{E}D_{E}[[g_{1}|g_{2}]] = 0. Part 2 follows.

3. Let x = i([g_{1}|g_{2}| ⋯ |g_{n}]) ∈ Σ^{−1}JP_{1}B[π], n > 2. (a)

D_{E}S_{I}[x] = D_{E}S_{I}[i([g_{1}|g_{2}| ⋯ |g_{n}])] = −D_{E}[s̃i([g_{1}|g_{2}| ⋯ |g_{n}])] = −D_{E}[is([g_{1}|g_{2}| ⋯ |g_{n}])] =

−D_{E} (∑_{g∈π}[i([g|g_{1}| ⋯ |g_{n}])]) =

∑_{g∈π} ∑_{k=1}^{n−2}(−1)^{k}[i([g|g_{1}| ⋯ |g_{k}])] · [i([g_{k+1}|g_{k+2}| ⋯ |g_{n}])]

22 4. HOMOLOGY EXPONENTS FOR ΩBπ_{p}^{∧}

(b)

S_{I}D_{E}[x] = S_{I}D_{E}[i([g_{1}|g_{2}| ⋯ |g_{n}])] =

S_{I} (∑_{k=2}^{n−2}(−1)^{k}[i([g_{1}|g_{2}| ⋯ |g_{k}])] · [i([g_{k+1}|g_{k+2}| ⋯ |g_{n}])]) =

∑_{k=2}^{n−2}(−1)^{k+1}[s̃i([g_{1}|g_{2}| ⋯ |g_{k}])] · [i([g_{k+1}|g_{k+2}| ⋯ |g_{n}])]

(c) Summing up the results of (a) and (b) above, we get:

(S_{I}D_{E} + D_{E}S_{I})[x] =

∑_{g∈π}[i[g|g_{1}|g_{2}]] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] −

∑_{g∈π}[i[g|g_{1}]] · [i([g_{2}|g_{3}| ⋯ |g_{n}])] −

[s̃i[g_{1}|g_{2}]] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] =

[is_{2}([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] − [is_{1}[g_{1}]] · [i([g_{2}|g_{3}| ⋯ |g_{n}])] − [s̃i([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])]

But

s̃i([g_{1}|g_{2}]) = is_{2}ηi([g_{1}|g_{2}]) =

is_{2}([g_{1}|g_{2}]) − is_{2}σd([g_{1}|g_{2}])

Thus

(S_{I}D_{E} + D_{E}S_{I})[x] = [is_{2}([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] − [is_{1}([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])] − [is_{2}([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] + [is_{2}σd([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] =

[is_{2}σd([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] − [is_{1}([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])]

(d) On the other hand

4.2. PROOF OF THEOREM 1.1.1 23

D_{I}S_{E}[x] = D_{I}S_{E}[i([g_{1}|g_{2}| ⋯ |g_{n}])] = D_{I}([is_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])]) = −{[dis_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])] + [is_{2}σ([g_{1}])] · [di([g_{2}|g_{3}| ⋯ |g_{n}])]}

(e)

S_{E}D_{I}[x] = −S_{E}[di([g_{1}|g_{2}| ⋯ |g_{n}])] = −S_{E}[id([g_{1}|g_{2}| ⋯ |g_{n}])] = −S_{E}[i([g_{2}|g_{3}| ⋯ |g_{n}])] +

∑_{k=1}^{n−1}(−1)^{k+1}S_{E}[i([g_{1}| ⋯ |g_{k}g_{k+1}| ⋯ |g_{n}])] +

(−1)^{n+1}S_{E}[i([g_{1}|g_{2}| ⋯ |g_{n−1}])] = −[is_{2}σ([g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] + [is_{2}σ([g_{1}g_{2}])] · [([g_{3}|g_{4}| ⋯ |g_{n}])] +

∑_{k=2}^{n−1}(−1)^{k+1}[is_{2}σ([g_{1}])] · [i([g_{2}| ⋯ |g_{k}g_{k+1}| ⋯ |g_{n}])] +

(−1)^{n+1}[is_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n−1}])] = −[is_{2}σ([g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] + [is_{2}σ([g_{1}g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] +

[is_{2}σ([g_{1}])] · ∑_{k=2}^{n−1}(−1)^{k+1}[i([g_{2}| ⋯ |g_{k}g_{k+1}| ⋯ |g_{n}])] +

(−1)^{n+1}[is_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n−1}])] = −[is_{2}σ([g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] + [is_{2}σ([g_{1}g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])] + [is_{2}σ([g_{1}])]([di([g_{2}|g_{3}| ⋯ |g_{n}])] − [i([g_{3}|g_{4}| ⋯ |g_{n}])]) = [is_{2}σ([g_{1}])] · [di([g_{2}|g_{3}| ⋯ |g_{n}])] − [is_{2}σd([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])]

(f) Summing up the results of (d) and (e) above, we get

24 4. HOMOLOGY EXPONENTS FOR ΩBπ_{p}^{∧}

(D_{I}S_{E} + S_{E}D_{I})[x] =

−{[dis_{2}σ([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])] + [is_{2}σd([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])]} But

dis_{2}σ([g_{1}]) = ids_{2}σ([g_{1}]) = i ∘ (ϕ_{π} − s_{1}d) ∘ σ([g_{1}]) = |π| · iσ([g_{1}]) − is_{1}dσ([g_{1}]) = −is_{1}([g_{1}])

Thus

(D_{I}S_{E} + S_{E}D_{I})[x] = [is_{1}([g_{1}])] · [i([g_{2}|g_{3}| ⋯ |g_{n}])] − [is_{2}σd([g_{1}|g_{2}])] · [i([g_{3}|g_{4}| ⋯ |g_{n}])]

Comparison of the results in (c) and (f) above proves part 3 of the proposition for n > 2. For n = 2, all the summands in 3 vanish. This completes the proof.

Under the assumptions of theorem 1.1.1 we have by proposition 4.2.1

(DS + SD)[x] = ϕ_{|π|}[x]

for any x ∈ Σ^{−1}J(P_{1}B[π]). By lemma 4.1.2 the theorem follows.

CHAPTER 5 Examples for Homology Exponents

In this chapter we present some examples for the behavior of ΩBπ_{p}^{∧} in terms of homology exponents. Recall that if π is a finite group, with a Sylow p-subgroup P, then the exponent of H̃_{∗}(Bπ; ℤ_{(p)}) is in fact bounded from above by that of H̃_{∗}(BP; ℤ_{(p)}). Thus let π be a finite p-perfect group, whose Sylow p-subgroup, P, has homology exponent p^{r} <| P |. The question naturally arising is, whether p^{r} · H̃_{∗}(ΩBπ_{p}^{∧}; ℤ_{(p)}) = 0. The answer to that turns out to be negative in general. We exhibit two examples to demonstrate that the torsion in H_{∗}(ΩBπ_{p}^{∧}; ℤ_{(p)}) is sometimes larger than the torsion in H_{∗}(Bπ; ℤ_{(p)}). We remark however that one can easily find examples of finite p-perfect groups π, such that, H̃_{∗}(Bπ_{p}^{∧}; ℤ_{(p)}) and H̃_{∗}(ΩBπ_{p}^{∧}; ℤ_{(p)}) share the same exponent, which is smaller than the order of the Sylow p-subgroup. For instance if π is a finite p-perfect group whose Sylow p-subgroup is cyclic of order p^{r}, then Bπ and ΩBπ_{p}^{∧} have the same p-primary homology exponent. Moreover, if π is isomorphic to a finite products of p-perfect groups of the form above, then Bπ and ΩBπ_{p}^{∧} have the same homology exponent, which is in general smaller than the order of the Sylow p-subgroup.

The computations presented here depend upon related facts, discussed in part 2 of this paper. We shall give explicit cross-references as we go along. Our examples are constructed at the prime 2. Thus in this chapter we write H^{∗}(−) for H^{∗}(−; 𝔽_{2}) and similarly for homology. Let S^{n}{k} denote the homotopy fibre of a degree k self-map of the sphere S^{n}.

5.1. Groups with a Dihedral Sylow 2-Subgroup

Recall that the dihedral group, D_{2^{n}}, of order 2^{n} is given by

D_{2^{n}} = {x, y | x^{2^{n−1}} = y^{2} = 1; yxy = x^{−1}}.

A faithful representation of D_{2^{n}} in the orthogonal group O(2) is given by

ρ(x) = ( \begin{matrix}cos(ζ_{n−1}) & −sin(ζ_{n−1}) \\ sin(ζ_{n−1}) & cos(ζ_{n−1}) \end{matrix} ) ρ(y) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

where ζ_{m} = 2π/2^{m}. The following lemma is well known.

Lemma 5.1.1. The torsion in H̃_{∗}(BD_{2^{n}}; ℤ_{(2)}) is bounded from above by 2^{n−1}

Proof. There is a map i : ℤ/2^{n−1}ℤ ⟶ D_{2^{n}} given by inclusion into the normal subgroup of D_{2^{n}}, generated by x. Write H^{∗}(Bℤ/2^{n−1}ℤ) ≅ P [u_{2}]⊗E[v_{1}] and H^{∗}(BD_{2^{n}}) ≅ P [x_{1}, y_{1}, w_{2}]/(x^{2}+xy), [26]. In computing i^{∗}, one immediately observes that i^{∗}(x) = v and i^{∗}(y) = 0. Furthermore, since w is the second Stiefel Whitney class of the representation ρ above, we have i^{∗}(w) = u.

Next, by [25], Sq^{1}(w) = wy. Thus the Bockstein spectral sequence on H^{∗}(BD_{2^{n}}) has the form:

E_{1} ≅ H^{∗}(BD_{2^{n}}; 𝔽_{2}); β_{1}x = x^{2}, β_{1}y = y^{2}, and β_{1}w = wy.

Thus E_{2} ≅ E[xw] ⊗ P [w^{2}]. Now, by the computation above, i^{∗}(xw) = vu. Thus by naturality, β_{j}(xw) = 0, for j ≤ n − 1 and β_{n−1}(xw) = w^{2}. Consequently, E_{n} ≅ E_{∞}. This completes the proof.

Let π be a finite 2-perfect group containing the dihedral group D_{2^{n}} as a Sylow 2-subgroup. By lemma 5.1.1, the torsion in H̃_{∗}(Bπ; ℤ_{(2)}) is bounded from above by 2^{n−1}. However we have

Proposition 5.1.1. Let π be a finite 2-perfect group containing the dihedral group D_{2^{n}} as a Sylow 2-subgroup. Then H̃_{∗}(ΩBπ_{2}^{∧}; ℤ_{(2)}) has 2^{n} torsion. In fact, for any k ≥ 1 there is an isomorphism, H_{2k}(ΩBπ_{2}^{∧}; ℤ_{(2)}) ≅ ℤ/2^{n}ℤ.

Proof. We compute the homology Bockstein spectral sequence for ΩBπ_{2}^{∧}. From [50, 41], it follows that π_{1}ΩBπ_{2}^{∧} ≅ ℤ/2ℤ. Thus by theorem 1.1.1 of part 2, there is a principal fibration of H-spaces and H-maps

S^{3}{2^{n+1}} ⟶ ΩBπ_{2}^{∧} ⟶ Bℤ/2ℤ. (5.1.1)

The ℤ_{(2)}-homology Serre spectral sequence for (5.1.1) has the form

E_{p,q}^{2} = H_{p}(Bℤ/2ℤ; H_{q}(S^{3}{2^{n+1}}; ℤ_{(2)})).

In particular, E_{0,2}^{2} = H_{2}(S^{3}{2^{n+1}}; ℤ_{(2)}) ≅ ℤ/2^{n+1}ℤ, is the unique non vanishing component of total dimension 2 in E_{∗,∗}^{2}. Thus H_{2}(ΩBπ_{2}^{∧}; ℤ_{(2)}) ≅ E_{0,2}^{∞}. By inspection of the spectral sequence one immediately observes that E_{0,2}^{∞} is either ℤ/2^{n+1}ℤ or a quotient of ℤ/2^{n+1}ℤ by ℤ/2ℤ. On the other hand, By theorem 1.1.1, 2^{n} · H_{2}(ΩBπ_{2}^{∧}; ℤ_{(2)}) = 0. Thus H_{2}(ΩBπ_{2}^{∧}; ℤ_{(2)}) ≅ ℤ/2^{n}ℤ.

Next, by proposition 4.1.5 of part 2, H_{∗}(ΩBπ_{2}^{∧}) ≅ E[y_{1}, y_{2}] ⊗ P [α_{2}] as an algebra and the primary Bockstein operators are given by

Sq_{∗}^{1}(y_{2}) = y_{1} and Sq_{∗}^{1}(α_{2}) = 0.

Thus the Bockstein spectral sequence has the form

E_{1} ≅ E[y_{1}, y_{2}] ⊗ P [α_{2}]; β^{1}y_{2} = y_{1}, β^{1}α_{2} = 0.
E_{2} = E_{3} = ⋯ = E_{n−1} ≅ E[y_{1}y_{2}] ⊗ P [α_{2}]; β^{i} = 0.
E_{n} ≅ E[y_{1}y_{2}] ⊗ P [α_{2}]; β^{n}(y_{1}y_{2}) = α_{2}.

This completes the proof.

Corollary 5.1.1. Let π be a finite 2-perfect group containing the dihedral group D_{2^{n}} as a Sylow 2-subgroup. Then H_{∗}(ΩBπ_{2}^{∧}; ℤ_{(2)}) ⊗ ℤ/2^{n}ℤ is periodic.

Proof. By proposition 5.1.1, we have

H_{p}(ΩBπ_{2}^{∧}; ℤ_{(2)}) ≅ \{ \begin{matrix}ℤ_{(2)} & p = 0 \\ ℤ/2ℤ & p ≡ 1 mod(2) \\ ℤ/2^{n}ℤ & p ≡ 0 mod(2). \end{matrix}

The result follows.

Notice in particular that ΩBπ_{2}^{∧} in proposition 5.1.1 is not simply connected. One might wonder if the proposition is true exactly because of this fact. The following example shows that this is not the case.

5.2. Groups with a Semidihedral Sylow 2-Subgroup

Recall that the semidihedral group SD_{2^{n}} is the group given by

SD_{2^{n}} = ⟨x, y|x^{2^{n−1}} = y^{2} = 1; yxy^{−1} = x^{2^{n−2}−1}⟩.

Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. The torsion in H̃_{∗}(BSD_{2^{n}}; ℤ_{(2)}) is known to be bounded by 2^{n−1} [25]. Thus the same is true for H̃_{∗}(BG; ℤ_{(2)}).

Proposition 5.2.1. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then H̃_{∗}(ΩBG_{2}^{∧}; ℤ_{(2)}) has 2^{n} torsion. In fact, for any k ≥ 1 there is an isomorphism, H_{4k}(ΩBG_{2}^{∧}; ℤ_{(2)}) ≅ ℤ/2^{n}ℤ.

Proof. As in proposition 5.1.1, we compute the homology Bockstein spectral sequence for ΩBG_{2}^{∧}. By theorem 1.1.2 of part 2, there is a principal fibration

S^{3}{2^{n−1}} ⟶ ΩBG_{2}^{∧} ⟶ F_{n} (5.2.1)

where F_{n} is a simply connected 2-local space with H^{∗}(F_{n}) ≅ H^{∗}(S^{5}{2}) as algebras over the Steenrod algebra.

The ℤ_{(2)}-homology Serre spectral sequence for (5.2.1) has the form

E_{p,q}^{2} = H_{p}(F_{n}; H_{q}(S^{3}{2^{n−1}}; ℤ_{(2)})).

In particular

E_{0,4}^{2} = H_{4}(S^{3}{2^{n−1}}; ℤ_{(2)}) ≅ ℤ/2^{n−1}ℤ and E_{4,0}^{2} = H_{4}(F_{n}; ℤ_{(2)}) ≅ ℤ/2ℤ

are the only non-vanishing components of total dimension 4 in E_{∗,∗}^{2}. By inspection of the spectral sequence, one immediately observes that E_{0,4}^{2} ≅ E_{0,4}^{∞}. Thus H_{4}(ΩBG_{2}^{∧}; ℤ_{(2)}) fits in an extension

0 ⟶ ℤ/2^{n−1}ℤ ⟶ H_{4}(ΩBG_{2}^{∧}; ℤ_{(2)}) ⟶ ℤ/2ℤ ⟶ 0 (5.2.2)

By proposition 4.2.6 of part 2, H_{∗}(ΩBG_{2}^{∧}) ≅ P [z_{4}] ⊗ E[u_{2}, v_{3}] as an algebra. Thus, in particular, H_{4}(ΩBG_{2}^{∧}) ≅ ℤ/2ℤ. Therefore the extension (5.2.2) is nonsplit and so H_{4}(ΩBG_{2}^{∧}; ℤ_{(2)}) ≅ ℤ/2^{n}ℤ. Also, since u_{2}v_{3} ∈ H_{5}(ΩBG_{2}^{∧}) is the unique 5-dimensional class, we conclude that the n-th Bockstein, β^{n}(u_{2}v_{3}) is defined and equal to z_{4}.

Next, let K < G be a 2-perfect subgroup containing the quaternion group Q_{2^{n−1}} as a Sylow 2-subgroup. The existence of such a subgroup is granted by proposition 4.2.4 of part 2. Also the natural map BK_{2}^{∧} ⟶ BG_{2}^{∧} induces an epimorphism on mod-2 cohomology (proposition 4.2.5, part 2). Thus, it follows that the (n − 1) Bockstein, β^{n−1}(v_{3}) is defined and equal to u_{2}. Therefore the homology Bockstein spectral sequence for ΩBG_{2}^{∧} is given as follows.

E^{1} = E^{2} = ⋯ E^{n−2} ≅ H_{∗}(ΩBG_{2}^{∧}); β^{i} = 0, i = 1 ⋯ n − 2.
E^{n−1} ≅ H_{∗}(ΩBG_{2}^{∧}); β^{n−1}(v_{3}) = u_{2}, β^{n−1}(z_{4}) = 0.
E^{n} ≅ P [z_{4}] ⊗ E[u_{2}v_{3}]; β^{n}(u_{2}v_{3}) = z_{4}.

This completes the proof.

Corollary 5.2.1. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then H_{∗}(ΩBG_{2}^{∧}; ℤ_{(2)}) ⊗ ℤ/2^{n}ℤ is periodic.

Proof. By proposition 5.2.1, we have

H_{p}(ΩBG_{2}^{∧}; ℤ_{(2)}) ≅ \{ \begin{matrix}ℤ_{(2)} & p = 0 \\ ℤ/2^{n−1}ℤ & p ≡ 2 mod(4) \\ ℤ/2^{n}ℤ & p ≡ 0 mod(4) \\ 0 & otherwise. \end{matrix}

The result follows.

30 5. EXAMPLES FOR HOMOLOGY EXPONENTS

CHAPTER 6 The Homotopy Groups of Bπ_{p}^{∧}

In this chapter we consider π_{∗}Bπ_{p}^{∧}. We start by proving theorem 1.1.4. Then in section 6.3 we present examples for the existence of unstable homotopy exponents for certain spaces of the form Bπ_{p}^{∧}. One might conjecture that Bπ_{p}^{∧} has a homotopy exponent for every finite group π.

6.1. Some Basic Facts

Spaces of type ΩBπ_{p}^{∧}, for π finite and p-perfect, have some general features, which appear rather unusual. One of these properties is the fact that all such spaces are quasibounded at the prime p.

Recall that a space X is said to be quasibounded at a prime p if for every positive integer n there exists α(n), such that the map A_{(p)}^{q} ⊗ H^{n}(X; 𝔽_{p}) ⟶ H^{n+q}(X; 𝔽_{p}) is zero for q ≥ α(n). Here A_{(p)}^{q} means all the elements in the mod-p Steenrod algebra of excess exactly q. Quasiboundness is a pleasant property to work with since one has

Theorem 6.1.1 (Lannes, Schwartz). Let p be a prime. Let f : E ⟶ B be a fibration with fibre F. Suppose that B is simply-connected and that both E and B are quasibounded at p. Then so is F.

The following proposition is an immediate generalization of a similar statement proven by F. Cohen in [17].

Proposition 6.1.1 (Cohen). Let π be a finite p-perfect group. Then ΩBπ_{p}^{∧} is quasibounded at p.

32 6. THE HOMOTOPY GROUPS OF Bπ_{p}^{∧}

Proof. Let ρ : π ⟶ SU(n) be a faithful representation. Identify π with its image in SU(n) via ρ. The group π acts properly on SU(n) by left translations, and the orbit space SU(n)/π is a finite dimensional manifold with fundamental group given by π. Since BSU(n) is simply-connected, the p-completion of the fibration

SU(n)/π ⟶ Bπ ⟶ BSU(n)

is also a fibration.

By pulling back twice we get a fibration

ΩBπ_{p}^{∧} ⟶ SU(n)_{p}^{∧} ⟶ (SU(n)/π)_{p}^{∧}

in which the base space is simply connected and both the base and the total spaces are finite dimensional. In particular they are quasibounded. Thus Theorem 6.1.1 applies to give the required result.

Next, recall that a group G is said to be locally finite if G is a direct limit of its finite subgroups. Miller’s theorem on the Sullivan conjecture [48] states the following:

Theorem 6.1.2 (Miller). Let G be a locally finite group. Let X be a space which is quasibounded at p. Then the mapping space Map_{∗}(BG_{p}^{∧}, X) is weakly contractible.

We now recall a few elementary facts on the Postnikov decomposition of a space. Let X be a space. The Postnikov tower associated with X is a tower of principal fibrations

⋯ ⟶^{p_{n}} X_{n} ⟶^{p_{n−1}} X_{n−1} ⟶^{p_{n−2}} · · · · · · ⟶^{p_{3}} X_{3} ⟶^{p_{2}} X_{2} ⟶^{p_{1}} X_{1}

with

1. X_{1} ≃ K(π_{1}X, 1).

2. The fibration X_{n} ⟶ X_{n−1} is induced from the path-loop fibration over K_{n} by a map k_{n} : X_{n−1} ⟶ K_{n}, where K_{n} ≃ K(π_{n}X, n + 1).

3. There is a map l_{n} : X ⟶ X_{n} for every n ≥ 1, such that p_{n−1}l_{n} = l_{n−1} and l_{n} induces an isomorphism on π_{q}(−) for 0 ≤ q ≤ n. The maps k_{n} are traditionally called the k-invariants.

We say that a space X is constructible by means of its Postnikov tower if the natural map

l : X ⟶ lim_{\overleftarrow{n}} X_{n}

is a weak equivalence. It is well known that nilpotent spaces are constructible by means of their Postnikov tower. If R is a Z-torsion-free principal ideal domain, then R-acyclic spaces associated to R-perfect spaces are also constructible by means of a Postnikov tower [22].

Through the end of section 6.2 we specialize to spaces which are constructible by means of their Postnikov tower. Thus by a space we mean, a space together with its Postnikov tower. By a GEM we mean a generalized Eilenberg-MacLane space, i.e. a product of Eilenberg-MacLane spaces.

Let X be a space. We say that X admits a finite Postnikov decomposition if X_{n} ≃ X_{n+1} for n sufficiently large. Thus X admits a finite Postnikov decomposition if and only if it has only finitely many non-vanishing homotopy groups. Suppose that in the Postnikov tower for a space X the n-th k-invariant k_{n} is null homotopic. Then X_{n+1} ≃ X_{n} × ΩK_{n}. Thus the Postnikov decomposition of a space has only finitely many non-vanishing k-invariants if and only if X ≃ X′ × K, where X′ admits a finite Postnikov decomposition and K is a GEM.

6.2. Proof of Theorem 1.1.4

Let π be a p-perfect group. Let R = 𝔽_{p} or ℤ_{(p)}. By proposition 1.1.2 one has a principal π-bundle:

γ : L^{R}π ⟶ A^{R}Bπ (6.2.1)

where L^{R}π ≃ ΩBπ_{p}^{∧} and A^{R}Bπ ≃ A^{p}π. Given a finite p-perfect group π, we consider the Postnikov tower associated with A^{p}π.

Proposition 6.2.1. Let π be a finite p-perfect group. Then A^{p}π has infinitely many non-vanishing k-invariants.

Proof. Assume the converse. Then A^{p}π ≃ A′ × K, where A′ admits a finite Postnikov decomposition and K is a GEM. Since H̃_{∗}(ΩBπ_{p}^{∧}; ℤ_{(p)}) is all torsion, so is π_{∗}(ΩBπ_{p}^{∧}). By the long exact homotopy sequence for (6.2.1) above, π_{n}A^{p}π is finite for all n. Thus π_{n}K is finite and p-local for all n. But a space of type K(G, n) with G finite and p-local always has non-vanishing mod-p homology. Thus we must have that K is trivial, so A^{p}π admits a finite Postnikov decomposition.

Let

⋯ ⟶ A_{n}^{p}π ⟶ A_{n−1}^{p}π ⟶ ⋯ A_{2}^{p}π ⟶ A_{1}^{p}π ⟶ Bπ

be the Postnikov tower for A^{p}π. Let n be the minimal integer such that A_{m}^{p}π ≃ A_{m+1}^{p}π , ∀m ≥ n. Then A^{p}π ≃ A_{n}^{p}π and we have a diagram:

[figure: ΩBπ_{p}^{∧}; K(G, n); A_{n}^{p}π; A_{n−1}^{p}π; ⋮; Bπ]

where G is isomorphic to the top non-vanishing homotopy group of A^{p}π. Since the following sequence

ΩBπ_{p}^{∧} ⟶ A_{n}^{p}π ⟶ Bπ

is a fibration and the composite

K(G, n) ⟶ A_{n}^{p}π ⟶ Bπ

is null homotopic, it follows that there is a map α : K(G, n) ⟶ ΩBπ_{p}^{∧}, inducing an isomorphism in π_{n}(−). But G is a finite p-local group and ΩBπ_{p}^{∧} is quasibounded at p. This contradicts theorem 6.1.2, and thus completes the proof.

Let π be a finite p-perfect group. By proposition 6.2.1, A^{p}π has infinitely many non-vanishing k-invariants. By proposition 1.1.2, ΩBπ_{p}^{∧} is homotopy equivalent to a finite covering of A^{p}π. Thus, ΩBπ_{p}^{∧} also has infinitely many non-vanishing k-invariants. This completes the proof of theorem 1.1.4.

6.3. Examples for Homotopy Exponents

In this section we study some examples where Bπ_{p}^{∧} has homotopy exponents, for π finite and p-perfect. Our examples depend upon the existence of spherical resolutions for some spaces of the form ΩBπ_{p}^{∧}, shown in part 2 of this work.

Let X be a space and C be a family of spaces. We say that X is finitely resolvable by fibrations over C if there exists a tower of principal fibrations:

X_{n} ⟶ X_{n−1} ⟶ ⋯ ⟶ X_{2} ⟶ X_{1} = X

such that X_{n} ∈ C and for each 1 ≤ i ≤ n, the fibration, X_{i+1} ⟶ X_{i} is induced from the path loop fibration over some B_{i} ∈ C via a map ρ_{i} : X_{i} ⟶ B_{i}.

In part 2 we give a systematic account of resolutions by fibrations over an arbitrary family of topological spaces. In particular let S be the family of all spheres and their iterated loop-spaces. We show the existence of finite S-resolutions for various spaces of type ΩBπ_{p}^{∧}, where π is a finite and p-perfect group. Since every space in S has homotopy exponents [20, 29, 59], we have:

Proposition 6.3.1. Let π be a finite p-perfect group such that ΩBπ_{p}^{∧} admits a finite S-resolution. Then Bπ_{p}^{∧} has a homotopy exponent.

Proof. This is just a restatement of proposition 3.1.2 in part 2 for C = S.

The particularly interesting examples are those where the exponent can be related to the order of the Sylow p-subgroup of π. Let S^{n}{k} denote the homotopy fibre of a degree-k self-map of the n-sphere. Recall from [20]:

Theorem 6.3.1 (Cohen, Moore Neisendorfer). Let p be an odd prime. Then S^{2n+1}{p^{r}} has a homotopy exponent p^{r}.

Proposition 6.3.2. Let p be an odd prime. Let π be a finite p-perfect group with a cyclic Sylow p-subgroup of order p^{r}. Then

p^{r} · π_{∗}Bπ_{p}^{∧} = 0.

Proof. In [35] and more systematically in [17] it is proven that in this case ΩBπ_{p}^{∧} ≃ S^{2n+1}{p^{r}}. Thus this is just a restatement of theorem 6.3.1.

Similarly one has in [17] that if π is a finite 2-perfect group whose Sylow 2subgroup is isomorphic with a generalized quaternion group Q_{2^{r}}, then ΩBπ_{2}^{∧} ≃ S^{3}{2^{r}}. It is again well known [20] that 2^{r} annihilates π_{∗}S^{3}{2^{r}} for r ≥ 2.

We point out that if π is a finite 2-perfect group containing a dihedral group D_{2^{n}} as a Sylow 2-subgroup, then the 2-connected cover of Bπ_{2}^{∧} is of the form Bπ̃_{2}^{∧}, where π̃ is a finite 2-perfect group containing a quaternion group Q_{2^{n+1}} as a Sylow 2-subgroup. Thus Bπ_{2}^{∧} has homotopy exponent 2^{n+1}.

The larger and possibly more interesting family of examples, where an exponent can be expressed in terms of the order of the Sylow p-subgroup is provided by finite perfect groups of classical Lie type.

Proposition 6.3.3. Let l be a prime number. Let q = l^{k} for some k. Let p be an odd prime divisor of (q^{2} −1). Let G_{n}(q) denote either the special linear group SL_{n}(𝔽_{q}) or the symplectic group Sp_{2n}(𝔽_{q}). Let P < G_{n}(q) denote the Sylow p-subgroup of G_{n}(q). Then the order of P annihilates the homotopy groups of BG_{n}(q)_{p}^{∧}

Proof. Let G_{n}(q) be the special linear group. For any integer m, let ν_{p}(m) denote the highest power of p dividing m. By theorem 1.1.4 of part 2, there is an equivalence ΩBG_{2}(q)_{p}^{∧} ≃ S^{3}{p^{r_{2}}} and for every n ≥ 3 there is a fibration:

ΩBG_{n−1}(q)_{p}^{∧} ⟶ ΩBG_{n}(q)_{p}^{∧} ⟶ S^{2n−1}{p^{r_{n}}}

where r_{j} = ν_{p}(q^{j} −1). By theorem 6.3.1 and proposition 3.1.2 in part 2, it follows that ΩBG_{n}(q)_{p}^{∧} has a homotopy exponent given by p^{N}, where N = ∑_{i=2}^{n} r_{i}. On the other hand the order of G_{n}(q) is q^{n} ∏_{i=2}^{n}(q^{i} − 1). But

ν_{p}(q^{n} ∏_{i=2}^{n}(q^{i} − 1)) = ν_{p}(∏_{i=2}^{n}(q^{i} − 1)) = ∑_{i=2}^{n} ν_{p}(q^{i} − 1) = ∑_{i=2}^{n} r_{i}.

This completes the proof for the special linear groups.

Similarly if G_{n}(q) is the symplectic group then by theorem 1.1.5 of part 2,

ΩBG_{1}(q)_{p}^{∧} ≃ S^{3}{p^{r_{2}}}

and for every n ≥ 2 there is a fibration:

ΩBG_{n−1}(q)_{p}^{∧} ⟶ ΩBG_{n}(q)_{p}^{∧} ⟶ S^{4n−1}{p^{s_{n}}}

where s_{n} = ν_{p}(q^{2n} − 1). The same argument as above gives that ΩBG_{n}(q)_{p}^{∧} has a homotopy exponent given by p^{N}, where N = ∑_{i=1}^{n} s_{i}. On the other hand, the order of G_{n}(q) in this case is q^{n^{2}} ∏_{j=1}^{n}(q^{2j} − 1). The result follows.

For further discussion of exponents in π_{∗}Bπ_{p}^{∧}, we refer the reader to chapter 7 of part 2.

CHAPTER 7 Stable Homotopy Exponents for ΩBπ_{p}^{∧}

In this chapter we prove theorem 1.1.5 and proposition 1.1.3. The proof of both statements makes use of some classical results from the theory of the stable transfer map for finite coverings. For the convenience of the reader, we include a review of those results in 7.1 below.

7.1. Preliminaries on the Transfer

The material presented here is well known. The reader is referred to [31] for complete details.

A little n-cube is an affine, orientation preserving embedding c :I^{n}^{∘}⟶I^{n}^{∘}, with edges parallel to the coordinate axes. Let C_{n}(k) denote the space of k little n-cubes [7], given by

C_{n}(k) = {(c_{1}, c_{2}, ⋯ c_{k}) | c_{i} = little n−cube; Im(c_{i}) ∩ Im(c_{j}) = ∅ if i ≠ j} .

The space C_{n}(k) is topologized as a subspace of the space of continuous maps

C : ∐_{k} I^{n}^{∘}⟶I^{n}^{∘}

given the compact-open topology.

The symmetric group on k letters, Σ_{k}, acts freely on C_{n}(k) from the right by permuting little cubes. There are inclusions σ_{n} : C_{n}(k) ↪ C_{n+1}(k), given by

σ_{n}(c_{1}, ⋯, c_{k}) = (c_{1} × 1, ⋯ c_{k} × 1)

where 1 = 1_{(0,1)}. Let C(k) = ⋃_{n} C_{n}(k). Then C(k) admits a free right Σ_{k} action and is known to be contractible [46].

Let G be a group. Let p : X ⟶ B be a principal G-bundle. Then X is a free G-space and p : X ⟶ B is isomorphic to

X ⟶ X/G. (7.1.1)

38 7. STABLE HOMOTOPY EXPONENTS FOR ΩBπ_{p}^{∧}

Equivalently, let EG be a free contractible right G-space. Let X be a left G-space and let G act diagonally on the product EG × X. Then (7.1.1) is isomorphic to the G-bundle: EG × X ⟶ EG ×_{G} X, i.e. the projection to the second factor EG × X ⟶ X induces a map of G-bundles:

[figure: EG × X; X; EG ×_{G} X; X/G]

where both horizontal arrows are homotopy equivalences.

Let G be a finite group of order k, with {τ_{1}, τ_{2}, ⋯ τ_{k}} a fixed ordering on the elements of G. The group G acts on itself by right translations as a permutation group. Let ρ : G ⟶ Σ_{k} be the corresponding faithful representation, given by the formula

τ_{ρ(g)(i)} = τ_{i}g; ∀g ∈ G.

For every n ≥ 1, one can use ρ to define a free right G action on C_{n}(k) by the formula

(c_{1}, c_{2}, ⋯ c_{k}) · g = (c_{ρ(g)(1)}, c_{ρ(g)(2)}, ⋯ c_{ρ(g)(k)}).

The action is compatible with the inclusions σ_{n} : C_{n}(k) ↪ C_{n+1}(k) and thus C(k) inherits the structure of a free contractible right G-space.

For any left G-space X, consider the left Σ_{k} action on X^{k}, given by permuting coordinates. The product C(k)×X^{k}, with a diagonal Σ_{k} action is a free Σ_{k} space. Define a map Φ : C(k) × X ⟶ C(k) × X^{k} by the formula

Φ((c_{1}, c_{2} ⋯ c_{k}); x) = ((c_{1}, c_{2} ⋯ c_{k}); τ_{1}x, τ_{2}x, ⋯, τ_{k}x).

By [31], Φ does not depend on the ordering chosen on G, up to homotopy, and is ρ-equivariant. Thus Φ induces a map of orbit spaces

Φ̄ : C(k) ×_{G} X ⟶ C(k) ×_{Σ_{k}} X^{k}.

Let (−)_{+} denote the addition of a disjoint base-point.

Definition 7.1.1 (Kahn-Priddy). Let X ⟶ X/G be a principal G-bundle. As before we identify the orbit space X/G with C(k)×_{G}X. Define the pretransfer, T : X/G ⟶ C(k) ×_{Σ_{k}} (X_{+})^{k} to be the composite

X/G ⟶^{Φ̄} C(k) ×_{Σ_{k}} X^{k} ⟶ C(k) ×_{Σ_{k}} (X_{+})^{k},

where the second map is induced by the inclusion X ↪ X_{+}.

Next, for any pointed connected space Y, let

QY = Ω^{∞}Σ^{∞}Y = colim_{\overrightarrow{n}} Ω^{n}Σ^{n}Y

and let E : Y ⟶ QY be the stable Freundental suspension.

The natural map given by 1 × E^{k} : C(k) × (X_{+})^{k} ⟶ C(k) × (Q(X_{+}))^{k} is Σ_{k}-equivariant. Thus it induces a map of orbit spaces

1 ×_{Σ_{k}} E^{k} : C(k) ×_{Σ_{k}} (X_{+})^{k} ⟶ C(k) ×_{Σ_{k}} (Q(X_{+}))^{k}.

For any pointed connected space Y, consider the iterated loop space Ω^{n}Y as the mapping space Map(I^{n}, İ^{n}; Y, y_{0}). There is a map θ_{n}^{k} : C_{n}(k)×_{Σ_{k}} (Ω^{n}Y)^{k} ⟶ Ω^{n}Y given by

θ_{n}^{k}((c_{1}, ⋯, c_{k}); (ω_{1}, ⋯, ω_{k}))(u) = \{ \begin{matrix}ω_{i}(c_{i}^{−1}(u)) & if u ∈ Im(c_{i}) \\ y_{0} & otherwise. \end{matrix}

Let Σ(Y) denote the suspension spectrum associated to Y.

Definition 7.1.2 (Kahn-Priddy). Let G be a finite group of order k. Let p : X ⟶ X/G be a principal G-bundle. Let t̂r : (C(k) ×_{G} X)_{+} ⟶ Q(X_{+}) denote the unique pointed extension of the composite

C(k) ×_{G} X ⟶^{T} C(k) ×_{Σ_{k}} (X_{+})^{k} ⟶^{1×_{Σ_{k}} E^{k}} C(k) ×_{Σ_{k}} (Q(X_{+}))^{k} ⟶^{θ_{∞}^{k}} Q(X_{+}).

The stable transfer map, tr : Σ((C(k) ×_{G} X)_{+}) ⟶ Σ((X_{+})) is defined to be the map whose stable adjoint is t̂r.

7.2. Proof of Theorem 1.1.5

We now turn back to spaces of type ΩBπ_{p}^{∧}. Let π be a finite p-perfect group of order k. Let

π = {τ_{1}, τ_{2}, ⋯ τ_{k}}

be some fixed ordering on π. Let ρ : π ⟶ Σ_{k} be the faithful representation resulting from the right action of π on itself. By proposition 1.1.2, there is a principal π-bundle

γ : L^{p}π ⟶ A^{p}π, (7.2.1)

where L^{p}π ≃ ΩBπ_{p}^{∧} and A^{p}π is p-acyclic. As in section 7.1, we consider C(k) as a free contractible right π-space, via the faithful representation ρ and identify (7.2.1) with:

β : C(k) × L^{p}π ⟶ C(k) ×_{π} L^{p}π (7.2.2)

We now study the stable transfer map associated to (7.2.2). First we need

Lemma 7.2.1. Let A be a connected p-acyclic space. Let L be a loop space on a p-local space. Then the mapping space Map_{∗}(A; L) is weakly contractible.

Proof. Write L = ΩL′. Then

(7.2.3) Map_{∗}(A; L) = Map_{∗}(A; ΩL′) ≃ Map_{∗}(ΣA; L’)

But ΣA is p-acyclic and simply connected. Thus ΣA_{(p)} is contractible. Let f : ΣA ⟶ L′ be any map. Since L′ is p-local, it follows that f factors through ΣA_{(p)} and thus is null homotopic. This shows that π_{0}(Map_{∗}(A; L)) = {1}. To show that the higher homotopy groups vanish observe that π_{n}(Map_{∗}(A; L)) ≅ π_{0}(Map_{∗}(Σ^{n}A; L)) = {1}.

As an immediate corollary of lemma 7.2.1, we get

Proposition 7.2.1. Let π be a finite p-perfect group. Then the stable transfer map for the principal π- bundle, γ : L^{p}π ⟶ A^{p}π, is null-homotopic.

Proof. By definition 7.1.2, the stable transfer map is the stable adjoint to t̂r : (C(k)×_{π} L^{p}π)_{+} ⟶ Q(L^{p}π_{+}), which in turn is the unique pointed extension of the composite

C(k) ×_{π} L^{p}π ⟶^{T} C(k) ×_{Σ_{k}} (L^{p}π_{+})^{k} ⟶^{1×_{Σ_{k}} E^{k}} C(k) ×_{Σ_{k}} (Q(L^{p}π_{+}))^{k} ⟶^{θ_{∞}^{k}} Q(L^{p}π_{+})

But C(k) ×_{π} L^{p}π ≃ A^{p}π is p-acyclic and Q(L^{p}π_{+}) is the loop space on a p-local space. The result follows immediately from lemma 7.2.1.

Next consider the map

Υ : C(k) × Q(L^{p}π_{+}) ⟶ Q(L^{p}π_{+})

given by the composite

C(k) × Q(L^{p}π_{+}) ⟶^{Φ} C(k) × (Q(L^{p}π_{+}))^{k} ⟶^{θ_{∞}^{k}∘β} Q(L^{p}π_{+})

where Φ and θ_{∞}^{k} are the maps defined in 7.1 above and β is the bundle projection.

Lemma 7.2.2. Let π be a finite p-perfect group of order k. Then

Υ : C(k) × Q(L^{p}π_{+}) ⟶ Q(L^{p}π_{+})

induces multiplication by k on homotopy groups Proof. The composite

C(k) × (Q(L^{p}π_{+}))^{k} ⟶^{β} C(k) ×_{Σ_{k}} (Q(L^{p}π_{+}))^{k} ⟶^{θ_{∞}^{k}} Q(L^{p}π_{+})

is equivalent to k-fold loop multiplication. Thus it induces k-fold addition on homotopy groups. Therefore it suffices to show that

C(k) × Q(L^{p}π_{+}) ⟶^{Φ} C(k) × (Q(L^{p}π_{+}))^{k}

induces the k-fold diagonal map.

By proposition 1.1.2, the action of π on L^{p}π by deck transformations induces the trivial action on homotopy groups. Thus it induces a trivial action on integral homology. Consider the Atiyah-Hirzebruch spectral sequence for π_{∗}^{s}L^{p}π:

E_{p,q}^{2} = H_{p}(L^{p}π; π_{∗}^{s}) ⟹ π_{∗}^{s}L^{p}π

It follows that π acts trivially on the E^{2} term of the spectral sequence. Thus the action on stable homotopy is trivial.

But

Φ = 1_{C(k)} × ((∏_{i=1}^{k} τ̄_{i}) ∘ Δ)

where τ_{i} run over the elements of π and τ̄_{i} = Q(τ_{i}). This completes the proof.

Having lemma 7.2.2, the proof of theorem 1.1.5 amounts to showing that Υ is null-homotopic. Indeed let k denote the order of π. Let C denote C(k) and let L denote L^{p}π. There is a commutative diagram:

[figure: C × L; Φ; β; C × Q(L_{+}); Υ; Φ; 1×(E∘i); C ×_{π} L; Φ̄; C × L^{k}; β; C × (Q(L_{+}))^{k}; β; 1×(E∘i)^{k}; C ×_{Σ_{k}} L^{k}; C ×_{Σ_{k}} (Q(L_{+}))^{k}; Q(L_{+}); 1×_{Σ_{k}} (E∘i)^{k}; θ]

where i : L ⟶ L_{+} is the inclusion.

Since t̂r : (C ×_{π} L)_{+} ⟶ Q(L_{+}) is null-homotopic, it follows, by commutativity, that the pointed extension of the composite:

C × L ⟶^{1×(E∘i)} C × Q(L_{+}) ⟶^{Υ} Q(L_{+})

is null-homotopic. Thus the stable adjoint of Υ ∘ (1 × (E ∘ i)) is null homotopic. But the stable adjoint of (1 × (E ∘ i))_{+} is an equivalence of spectra and Υ is by construction an infinite loop map. Thus Υ is null-homotopic. This completes the proof.

7.3. The Non-Existence of Exponents in π_{∗}^{s}Bπ_{p}^{∧}

Let Σ_{n} denote the symmetric group on n letters. Let Σ_{∞} = colim_{\overrightarrow{n}} Σ_{n} denote

the infinite symmetric group. There is a well known homotopy equivalence: φ_{∞} : BΣ_{∞}^{∧} ⟶ Q_{0}S^{0} [4], [32]. Recall from [31]:

Theorem 7.3.1 (Kahn-Priddy). Let p be a prime. Then the natural map ϕ : Q(BΣ_{p})_{p}^{∧} ⟶ (BΣ_{∞})_{p}^{∧} induces a split epimorphism on homotopy groups. Furthermore, the same is true for the composite:

QBℤ/pℤ ↪^{i}Q(BΣ_{p})_{p}^{∧} ⟶^{ϕ} (BΣ_{∞})_{p}^{∧}

where i is induced by the inclusion into a Sylow p-subgroup.

It is well known [39] that π_{∗}(BΣ_{∞})_{p}^{∧} has torsion of arbitrarily high order. Thus we get:

Corollary 7.3.1. Let p be a prime. Then π_{∗}^{s}(BΣ_{p})_{p}^{∧} and π_{∗}^{s}Bℤ/pℤ have torsion of arbitrarily high order.

Proposition 1.1.3 is merely a generalization of corollary 7.3.1. We start by making a few observations.

Lemma 7.3.1. Let G be a finite group of order n. Let H < G be a subgroup of order k and index m. Then there is a commutative diagram:

[figure: H; ρ^{H}; G; ρ^{G}; inc; Σ_{k}; Σ_{n}; θ]

where ρ^{H} and ρ^{G} are permutation representations of H and G respectively and θ is the composite:

Σ_{k} ⟶^{Δ} (Σ_{k})^{×m} ⟶^{ω} Σ_{n}

with Δ being the diagonal and ω the natural inclusion.

7.3. THE NON-EXISTENCE OF EXPONENTS IN π_{∗}^{s}Bπ_{p}^{∧} 43

Proof. Let h_{1}, h_{2}, ⋯, h_{k} denote a fixed ordering on the elements of H. Let g_{1}, g_{2}, ⋯, g_{m} be ordered representatives for the right cosets of H in G. Then the elements of G can be listed as follows:

h_{1}g_{1}, h_{2}g_{1}, ⋯, h_{k}g_{1}, h_{1}g_{2}, h_{2}g_{2}, ⋯, h_{k}g_{2}, · · · · · · h_{1}g_{m}, h_{2}g_{m}, ⋯, h_{k}g_{m}

Let ρ^{G} : G ⟶ Σ_{n} be the permutation representation corresponding to the left G action on itself with respect to the ordering above. Let ρ^{H} : H ⟶ Σ_{k} be the permutation representation corresponding to the left H action on itself with respect to the ordering on H. Clearly, for every h ∈ H, ρ^{G}(h) is a permutation which operates on each block:

h_{1}g_{i}, h_{2}g_{i}, ⋯, h_{k}g_{i}

the same way ρ^{H}(h) operates on h_{1}, h_{2}, ⋯, h_{k}. The result follows.

Let α : ℕ ⟶ ℕ be an order preserving monomorphism of sets. Let N(α) denote the image of α. Then α : ℕ ⟶ N(α) is an isomorphism of ordered sets. Define a function F_{α} : Σ_{∞} ⟶ Σ_{∞} by:

F_{α}(σ)(i) = \{ \begin{matrix}ασα^{−1}(i) & i ∈ N(α) \\ i & otherwise \end{matrix}

Clearly F_{α}(e) = e and if σ, τ ∈ Σ_{∞} and i ∈ N(α), we have

F_{α}(στ)(i) = αστα^{−1}(i) = ασα^{−1}ατα^{−1}(i) = F_{α}(σ)F_{α}(τ)(i).

Thus F_{α} is a group homomorphism. Notice that F_{α} can be considered as the inclusion into the subgroup of Σ_{∞}, which fixes ℕ \ N(α).

Lemma 7.3.2. Let α : ℕ ⟶ ℕ be an order preserving monomorphism of sets. Then F_{α} : Σ_{∞} ⟶ Σ_{∞} induces a homology isomorphism with respect to any coefficient module.

Proof. Let i_{n} : Σ_{n} ⟶ Σ_{∞} be the natural inclusion into the subgroup, which fixes the letter j, if j > n. Denote F_{α,n} = F_{α} ∘ i_{n}. Then we have a commutative diagram:

[figure: ⋯; Σ_{n−1}; F_{α,n−1}; Σ_{n}; F_{α,n}; Σ_{n+1}; F_{α,n+1}; ⋯; ⋯; Σ_{∞}; Σ_{∞}; Σ_{∞}; ⋯; =; =; =; =]

Thus one might think of F_{α} as the map induced on the colimit of the top row by the maps F_{α,n}. Since homology commutes with colimits of this type, it suffices to show that for all n ≥ 1 and any Σ_{n}-module M, H_{∗}(F_{α,n}; M) = H_{∗}(i_{n}; M). Let n ∈ ℕ. Let L(n) denote the cardinality of the set {1 ⋯ n} ∩ N(α). Let β_{n} : {1 ⋯ n} \ N(α) ⟶ ℕ be the function which assigns the natural linear order on {1 ⋯ n}\N(α). Notice that β_{n} has an inverse on the subset {1, ⋯, n−L(n)} ⊂ ℕ. Define an element τ_{n} = τ_{α,n} ∈ Σ_{n} by:

τ_{n}(i) = \{ \begin{matrix}α^{−1}(i) & i ∈ N(α) ∩ \{1 ⋯ n\} \\ L(n) + β_{n}(i) & otherwise. \end{matrix}

One routinely verifies that τ_{n}^{−1} is given by:

τ_{n}^{−1}(i) = \{ \begin{matrix}α(i) & 1 ≤ i ≤ L(n) \\ β_{n}^{−1}(i − L(n)) & L(n) < i ≤ n \end{matrix}

Identify Σ_{n} with i_{n}(Σ_{n}) < Σ_{∞}. Let m ∈ ℕ be fixed. Since N(α) is an infinite subset of ℕ, there exists k ≫ 0 so that L(k) > m. Thus for every σ ∈ Σ_{m} and i ∈ ℕ, we have:

τ_{k}^{−1}στ_{k}(i) = \{ \begin{matrix}τ_{k}^{−1}σα^{−1}(i) & i ∈ N(α) ∩ \{1 ⋯ k\} \\ τ_{k}^{−1}σ(L(k) + β_{k}(i)) & otherwise \end{matrix}

But α^{−1}(i) ≤ L(k) for i ∈ N(α) ∩ {1 ⋯ k}. Thus σα^{−1}(i) ≤ L(k) and so τ_{k}^{−1}σα^{−1}(i) = ασα^{−1}(i). On the other hand L(k) > m so τ_{k}^{−1}σ(L(k) + β_{k}(i)) = τ_{k}^{−1}(L(k) + β_{k}(i)) = β_{k}^{−1}(L(k) + β_{k}(i) − L(k)) = i. Hence τ_{k}^{−1}στ_{k} = F_{α}(σ) for every σ ∈ Σ_{m}. Consequently we get commutative diagrams:

[figure: Σ_{m}; F_{α,m}; Σ_{∞}; c_{τ_{k}}; i_{m}; Σ_{∞}; Σ_{∞}; =]

where c_{τ_{k}} is conjugation by τ_{k}^{−1}. Applying homology with coefficients in M and using the fact that inner automorphisms operate trivially on group homology, we get the required result.

Next we use lemma 7.3.2 in order to show that certain maps

μ_{m,k} : ((BΣ_{∞})^{∧})^{×m} ⟶ (BΣ_{∞})^{∧}

operate as m-fold multiplication maps. The idea for the construction is directly related to the classical construction of the standard H-space structure map for (BΣ_{∞})^{∧} (see for example [2, pp. 87]).

Let m, k be positive integers. Divide the set of natural numbers, ℕ, into m distinct subsets N_{k}(1), N_{k}(2), ⋯, N_{k}(m) by:

N_{k}(i) = {j ∈ ℕ | j ≡ (i − 1)k + 1, ⋯, ik − 1 or ik mod(mk)}

For example if m = 2 and k = 1, we have, N_{1}(1) = positive odd integers and N_{1}(2) = positive even integers. Let α_{i} : ℕ ⟶ N_{k}(i) be the map which embeds ℕ onto N_{k}(i) so that α_{i}(j) = the j-th point in N_{k}(i). The inverse α_{i}^{−1} : N_{k}(i) ⟶ ℕ is the map which assigns to each point its ordering index in N_{k}(i). Define a map η_{m,k} : Σ_{∞}^{×m} ⟶ Σ_{∞} by

η_{m,k}(σ_{1}, σ_{2}, ⋯, σ_{m})(k) = α_{i}σ_{i}α_{i}^{−1}(k), if k ∈ N_{k}(i)

In other words η_{m,k}(σ_{1}, σ_{2}, ⋯, σ_{m}) is the permutation which operates on N_{k}(i) by σ_{i}. In particular η_{m,k}(e, e, ⋯ e) = e and for l ∈ N_{k}(i), we have:

η_{m,k}((σ_{1}, ⋯, σ_{m}) · (τ_{1}, ⋯, τ_{m}))(l) =
η_{m,k}((σ_{1}τ_{1}, ⋯, σ_{m}τ_{m}))(l) =
α_{i}σ_{i}τ_{i}α_{i}^{−1}(l) =
α_{i}σ_{i}α_{i}^{−1}α_{i}τ_{i}α_{i}^{−1}(l) =
(η_{m,k}(σ_{1}, ⋯, σ_{m}) · η_{m,k}(τ_{1}, ⋯, τ_{m}))(l)

Thus η_{m,k} is a group homomorphism. Let μ_{m,k} = (B(η_{m,k}))^{∧}. Since both the classifying functor and the R-completion functor commute with products, we can consider μ_{m,k} as a map defined on ((BΣ_{∞})^{∧})^{×m}.

Let γ_{i} : Σ_{∞} ⟶ (Σ_{∞})^{×m} be the inclusion into the i-th factor. Then the map f_{m,k,i} = η_{m,k} ∘ γ_{i} : Σ_{∞} ⟶ Σ_{∞} is equal to F_{α_{i}} of lemma 7.3.2, i.e. it is the natural inclusion onto the subgroup of Σ_{∞} which fixes the subset ℕ \ N_{k}(i). Thus we get:

Corollary 7.3.2. For every m, k, i ∈ ℕ, the homomorphism f_{m,k,i} : Σ_{∞} ⟶ Σ_{∞} induces the identity map on homology with any coefficient system. Thus μ_{m,k} : ((BΣ_{∞})^{∧})^{×m} ⟶ (BΣ_{∞})^{∧} is an m-fold multiplication map.

Notice that the same result is true if one replaces the ℤ-completion functor by the p-completion functor. Also, as an immediate corollary of our construction we get:

Lemma 7.3.3. Let ω : Σ_{p}^{×m} ⟶ Σ_{mp} be the natural inclusion. Then with the notation above we have a commutative diagram:

[figure: Σ_{p}^{×m}; i_{p}^{×m}; Σ_{mp}; i_{mp}; ω; Σ_{∞}^{×m}; Σ_{∞}; η_{m,p}]

We are now ready to prove proposition 1.1.3. Let G be a finite group of order divisible by p. Let C < G be a cyclic subgroup of order p. Suppose |G : C| = m. By 7.3.1 and 7.3.3 there is a commutative diagram

[figure: BC; Bρ^{C}; (BG)_{p}^{∧}; Bρ^{G}; Bi; (BΣ_{p})_{p}^{∧}; Bi_{p}; ((BΣ_{p})_{p}^{∧})^{×m}; Bi_{p}^{×m}; (BΣ_{mp})_{p}^{∧}; Bi_{mp}; Δ; Bω; (BΣ_{∞})_{p}^{∧}; ((BΣ_{∞})_{p}^{∧})^{×m}; (BΣ_{∞})_{p}^{∧}; Δ; μ_{m,p}]

Notice that θ = ω ∘ Δ. By corollary 7.3.2 and commutativity,

Bi_{mp} ∘ Bθ = Bi_{m,p} ∘ Bω ∘ Δ = μ_{m,p} ∘ Δ ∘ Bi_{p} = m · Bi_{p}.

Let

ϕ : QBC ⟶ (BΣ_{∞})_{p}^{∧}

be the p-completed extension of B(i_{mp} ∘ θ ∘ ρ^{C}) : BC ⟶ (BΣ_{∞})_{p}^{∧} to QBC. Then ϕ = m · ψ, where ψ : QBC ⟶ (BΣ_{∞})_{p}^{∧} is induced by the composite i_{p} ∘ ρ^{C} : C ⟶ Σ_{∞}.

By theorem 7.3.1, ψ_{∗} : π_{∗}QBC ⟶ π_{∗}(BΣ_{∞})_{p}^{∧} is a split epimorphism. Let ε be a right inverse for ψ_{∗}. Then ϕ_{∗}ε : π_{∗}(BΣ_{∞})_{p}^{∧} ⟶ π_{∗}(BΣ_{∞})_{p}^{∧} is equal to multiplication by m.

Now let r be a non-negative integer. We show that π_{∗}^{s}BG_{p}^{∧} has an element of order at least p^{r+1}. Since π_{∗}(BΣ_{∞})_{p}^{∧} has elements of arbitrarily high order, there exists an element x ∈ π_{∗}(BΣ_{∞})_{p}^{∧} such that mp^{r}x = ϕ_{∗}ε(p^{r}x) ≠ 0. Thus QBi ∘ ε(x) ∈ π_{∗}^{s}BG_{p}^{∧} is an element of order at least p^{r+1}. This completes the proof.

Part 2

Finite Groups and Resolutions by Fibrations

48

CHAPTER 1 Introduction

In [17], F. Cohen gives examples of loop spaces of the form ΩBπ^{+}, which are finitely resolvable by fibrations over spheres, where π is a finite perfect group and (−)^{+} denotes the Quillen “plus” construction. Cohen’s conjecture stated in [17] suggests that such a resolution exists for ΩBπ^{+} for any finite perfect group π such that H_{2}(Bπ; ℤ) vanishes as well. As in part 1 it seems natural to work in the contexts of p-perfect groups rather than perfect groups and relate to a mod-p version on Cohen’s conjecture which suggests the same statement about ΩBπ_{p}^{∧} for any finite p-perfect group π with H_{2}(Bπ; ℤ_{(p)}) = 0.

The purpose of this part of the paper we produce further examples for the mod-p Cohen conjecture. A remark is due here. This work was completed by the end of 1992. Since than a counter example to the mod-p Cohen conjecture was discovered [37]. However we point out that (1) there is a sufficient number of examples to make one wonder whether there are some general phenomena to be explored here, (2) as we explain in chapter 7 below, each positive example for Cohen’s conjecture provides evidence for the Moore finite exponent conjecture and (3) the counter example presented in [37] does not rule out the possibility that a suitable modification of the mod-p Cohen conjecture might hold. Thus the author maintains the belief that the presentation of those examples is justified although the motivating conjecture turns out to be false as stated.

Throughout this part of the paper a space X shall mean, as always, a connected simplicial set or a CW complex. Let S denote the family of all spheres S^{n} and iterated loop spaces Ω^{n}S^{n+k}. The notion “S-resolution” is used in the statement of our result and will be made precise in chapter 3 below. We denote by S^{n}{p^{r}} the homotopy fibre of a degree-p^{r} self map of the n-sphere S^{n}. Let X be a simply-connected p-local space. We say that X is a space of type S^{n}{p^{r}} if H^{∗}(X; 𝔽_{p}) is isomorphic to H^{∗}(S^{n}{p^{r}}; 𝔽_{p}) as algebras over the mod-p Steenrod algebra and has the same Bockstein operators.

1.1. Statement of Results

We exhibit three different sets of finite p-perfect groups π, for which ΩBπ_{p}^{∧} is spherically resolvable. The first set of examples is of a sporadic nature. We use the (known) algebra structure of H^{∗}(Bπ; 𝔽_{2}) for certain finite 2-perfect groups together with standard machinery of homological algebra and the useful theory of [30] in order to derive certain properties of such classifying spaces after 2completion.

Theorem 1.1.1. Let π be a finite 2-perfect group containing the dihedral group, D_{2^{n}} as a Sylow 2-subgroup. Then the 1-connected cover of ΩBπ_{2}^{∧} is homotopy equivalent to S^{3}{2^{n+1}}.

Theorem 1.1.2. Let G be a finite 2-perfect group containing the semi-dihedral group, SD_{2^{n}} as a Sylow 2-subgroup. Then there is a fibration

S^{3}{2^{n−1}} ⟶ ΩBG_{2}^{∧} ⟶ F_{n}

where F_{n} is a space of type S^{5}{2}.

In all known examples for theorem 1.1.2, F_{n} ≃ aprioriS^{5}{2}, but a priori it is not clear that this is the case in general. Along the lines of proving theorems 1.1.1 and 1.1.2, we also compute the homology of ΩBπ_{2}^{∧} and ΩBG_{2}^{∧} with 𝔽_{2} coefficients as a Hopf algebra. The computations might have some interest in their own right. In particular, we show that these homology rings are commutative. The computation is used in chapter 5 of part 1, where we exhibit examples for the behavior of H_{∗}(ΩBπ_{2}^{∧}; ℤ_{(2)}) in terms of homology exponents. We also refer the reader back to corollaries 5.1.1 and 5.2.1 of part 1 to inspect certain periodicity phenomena in the homology of ΩBπ_{2}^{∧} and ΩBG_{2}^{∧}, where π and G are as in theorems 1.1.1 and 1.1.2.

Next we study finite groups of Lie type. We use the theory of E. Friedlander [27] in order to produce spherical resolutions for ΩBG_{p}^{∧} where G is a group of Lie type over a finite field of characteristic l and p is an odd prime different from l. Weaker results are also obtained at the prime 2.

The finite groups of classical Lie types A_{n} and C_{n} are the ones for which we obtain the most explicit results. Indeed if G is a finite group of Lie type A_{n} or C_{n}, we produce a resolution for ΩBG_{p}^{∧} in terms of spaces of the form S^{2n−1}{p^{r}}, for p odd, and spaces of type S^{2n−1}{2^{r}} if p = 2. In particular one is able to deduce the existence of homotopy exponents for BG_{p}^{∧} in those cases (see section 6.3 of part 1).

For a prime p and an integer m let ν_{p}(m) denote the highest power of p dividing m. The terminology used in theorem 1.1.3 below will be explained in chapter 5.

Theorem 1.1.3. Let G be a complex connected reductive Lie group which is algebraically resolvable at the prime p, over k odd spheres of dimensions 2m_{i} + 1, i = 1 ⋯ k. Let G_{1}, ⋯, G_{k} be a sequence of subgroups of G associated with the resolution. Let q = l^{n} for some prime l ≠ p and let G_{i}(𝔽_{q}) be the associated finite Chevalley groups. Let t_{i} = ν_{p}(s_{i} − 1), where s_{i} is the degree of the map induced on S^{2m_{i}+1} by ϕ^{q}. Then the groups G_{i}(𝔽_{q}) are p-perfect and for every i = 1 ⋯ k there are fibrations :

ΩBG_{i−1}(𝔽_{q})_{p}^{∧} ⟶ ΩBG_{i}(𝔽_{q})_{p}^{∧} ⟶ S_{i}

where S_{i} ≃ S^{2m_{i}+1}{p^{t_{i}}} if p is odd and S_{i} is a space of type S^{2m_{i}+1}{2^{t_{i}}} if p = 2. Thus for every 1 ≤ i ≤ k, the space ΩBG_{i}(𝔽_{q})_{p}^{∧} is finitely S-resolvable.

As particular examples for theorem 1.1.3, we have:

Theorem 1.1.4. Let p and q be as in theorem 1.1.3. Let SL_{n}(𝔽_{q}) denote the special linear group of rank n over 𝔽_{q}. Then for every n ≥ 2 there are fibrations

ΩBSL_{n−1}(𝔽_{q})_{p}^{∧} ⟶ ΩBSL_{n}(𝔽_{q})_{p}^{∧} ⟶ S_{n}

where S_{n} ≃ S^{2n−1}{p^{r_{n}}} if p is odd. If p = 2 then S_{n} is a space of type S^{2n−1}{2^{r_{n}}}. In either case, r_{n} = ν_{p}(q^{n} − 1). In particular for any n ≥ 2, the space ΩBSL_{n}(𝔽_{q})_{p}^{∧} is finitely S-resolvable.

Theorem 1.1.5. Let p and q be as in theorem 1.1.3. Let Sp_{2n}(𝔽_{q}) denote the symplectic group of rank 2n over 𝔽_{q}. Then for every n ≥ 1, there are fibrations

ΩBSp_{2n−2}(𝔽_{q})_{p}^{∧} ⟶ ΩBSp_{2n}(𝔽_{q})_{p}^{∧} ⟶ S_{n}

where S_{n} ≃ S^{4n−1}{p^{r_{n}}} if p is odd. If p = 2 then S_{n} is a space of type S^{4n−1}{2^{r_{n}}}. In either case, r_{n} = ν_{p}(q^{2n} − 1). In particular for any n ≥ 1, the space ΩBSp_{2n}(𝔽_{q})_{p}^{∧} is finitely S-resolvable.

The finite classical groups Spin_{n}(𝔽_{q}) and the finite groups of exceptional Lie type are harder to handle using our methods. However restricted results can still be obtained. Indeed, for any finite group G(q) of Lie type, whose corresponding complex Lie group G is reductive and p-perfect , we use [27] together with Borel’s structure theorem [8] on the cohomology of Lie groups and the theory of [21] in order to prove the existence of finite S-resolutions for ΩBG(q)_{p}^{∧} at most primes p.

Theorem 1.1.6. Let q = l^{k} be a prime power. Let G be a complex connected reductive Lie group. Let p be a prime different from l and suppose that G is p-perfect as a topological space and that H^{∗}(G; ℤ) has no p-torsion. Suppose further that

rank(H^{∗}(G; ℚ)) = n < p − 1.

Let G(q) be the associated finite group over the field 𝔽_{q}. Then ΩBG(q)_{p}^{∧} is finitely resolvable by fibrations over S.

The third set of examples that we produce, arises from the well known construction of Clark and Ewing [16] of spaces with polynomial mod-p cohomology algebra. These spaces, or rather their construction, are naturally related to finite groups whose Sylow p-subgroups are abelian. In particular if the Sylow p-subgroup P of a finite group π is isomorphic to ℤ/p^{r}ℤ^{×n} then we construct a map from Bπ_{p}^{∧} to a simply-connected p-complete space, which is in some sense a cohomological torsion free analogue of Bπ_{p}^{∧}. Furthermore, if the operation of the Weyl group W_{π}(P) ≅ N_{π}(P)/C_{π}(P) on P and the prime p satisfy certain conditions, then we are able to conclude that ΩBπ_{p}^{∧} is S-resolvable. The reader is referred to chapter 6 below for the specific results as there are too many details to be dealt with within the scope of an introduction.

1.2. Organization of Part 2

This part of the paper is organized as follows: Chapters 2 and 3 are preliminary in nature. In chapter 2 we discuss some basic concepts for reference through the rest of the paper. Chapter 3 includes a systematic account of resolutions by fibrations over an arbitrary family of topological spaces. Chapter 4 contains our sporadic examples of spherically resolvable spaces of the form ΩBπ_{p}^{∧}. We prove theorems 1.1.1 and 1.1.2 and calculate the homology rings of the resulting loop-spaces. In chapter 5 we discuss finite groups of Lie type and prove theorems 1.1.3, 1.1.4, 1.1.5 and 1.1.6. Chapter 6 contains a discussion of Clark-Ewing spaces and the associated classifying spaces of finite p-perfect groups. We conclude the paper with chapter 7 in discussing various modifications of the Cohen’s conjecture [17] which initiated this work. We also remark on some questions considering ΩBπ_{p}^{∧} that remain open.

CHAPTER 2 Preliminaries

2.1. Universal Central Extensions

Let R be a commutative ring with a unit. Recall that a group π is said to be R-prefect if H_{1}(Bπ; R) = 0.

Definition 2.1.1. Let R be a commutative ring with a unit. Let π be an R-perfect group. The R-universal central extension of π is the extension of π by H_{2} = H_{2}(Bπ; R) :

0 ⟶ H_{2} ⟶ π̃^{R} ⟶ π ⟶ 1

corresponding to the fundamental class in H^{2}(Bπ; H_{2}(Bπ; R)).

For R = ℤ or ℤ_{(p)} we denote π̃^{R} by π̃ or π̃^{p} respectively. The group π̃ is said to be the universal central extension of π and similarly the group π̃^{p} is called the p-universal central extension of π. Notice that a group π admits an R-universal central extension if and only if it is R-perfect.

A group π is said to be R-superperfect (superperfect if R = ℤ and psuperperfect if R = ℤ_{(p)}) if it is R-perfect and, in addition, H_{2}(Bπ; R) = 0. Thus π is R-superperfect if and only if π = π̃^{R}. The following proposition is evident.

Proposition 2.1.1. Let π be an R-perfect group. Then (Bπ̃^{R})_{R}^{∧} is the 2connected cover for Bπ_{R}^{∧}.

2.2. Uniqueness of Homotopy Type, Special Case

Notation 2.2.1. Let G be a group and H a subgroup. Let N_{G}(H) denote the normalizer of H in G. Let C_{G}(H) denote the centralizer of H in G. If G is finite and P < G is a Sylow p-subgroup, then the Weyl group W_{G}(P) is defined to be the quotient N_{G}(P)/P · C_{G}(P).

Let G, H be finite groups with isomorphic Sylow p-subgroup. Recently J. Martino and S. Priddy [43] introduced a criterion for deciding whether BG_{p}^{∧} and BH_{p}^{∧} are homotopy equivalent. Here we recall some elementary facts on the simplest possible case, i.e. the case where the Sylow p-subgroup is abelian. Our treatment is independent of the Martino-Priddy techniques.

By a theorem of R. Swan [58], if G is a finite group with an abelian Sylow p-subgroup P, then the inclusion N_{G}(P) ↪ G induces a homotopy equivalence

BN_{G}(P)_{p}^{∧} ⟶^{≃} BG_{p}^{∧}.

Let G be a finite group with an abelian normal Sylow p-subgroup P, then G ≅ P ⋊ G/P, where G/P acts on P by conjugations. Since P < C_{G}(P) and C_{G}(P) acts trivially on P it follows that the maps

P ⋊ (W_{G}(P)) ⟵ P ⋊ (N_{G}(P)/P) ≅ N_{G}(P) ⟶ G

induce mod-p cohomology isomorphism.

Corollary 2.2.1. Let G be a finite group with an abelian Sylow p-subgroup P. Then

B(P ⋊ W_{G}(P))_{p}^{∧} ≃ BG_{p}^{∧}

Proposition 2.2.1. Let G_{1} and G_{2} be finite groups with the same abelian Sylow p-subgroup P. Suppose that W_{G_{1}}(P) and W_{G_{2}}(P) are conjugate subgroups of Aut(P). Then

(BG_{1})_{p}^{∧} ≃ (BG_{2})_{p}^{∧}

Proof. Any inner automorphism of Aut(P) which sends W_{G_{1}}(P) to W_{G_{2}}(P) induces an isomorphism

P ⋊ W_{G_{1}}(P) ⟶ P ⋊ W_{G_{2}}(P)

The result follows immediately from corollary 2.2.1.

Example 2.2.1. Let T be an elementary abelian 2-group of rank 2. An example for a 2-perfect group containing T as a Sylow 2-subgroup is given by the alternating group on five letters A_{5}. Let G be any finite group containing T as a Sylow 2-subgroup. By Corollary 2.2.1, BG_{2}^{∧} ≃ B(T × W_{G}T)_{2}^{∧}. The automorphism group Aut(T) is isomorphic to GL_{2}(𝔽_{2}) ≅ Σ_{3}. Furthermore, the order of W_{G}T is prime to 2. Thus either W_{G}T ≅ {1} or W_{G}T ≅ ℤ/3ℤ, which has a unique representation in Σ_{3}, up to conjugacy. Hence either BG_{2}^{∧} ≃ BT or BG_{2}^{∧} ≃ (BA_{5})_{2}^{∧}.

2.3. Homotopy decomposition of Classifying Spaces

The material presented here is due to S. Jackowski and J. McClure [30] and will be used in chapter 4 below.

Definition 2.3.1. Let G be a compact Lie group. The Quillen category for G, denoted A_{p}(G), is the small category whose objects are all non-trivial elementary abelian p-subgroups of G. Morphisms in A_{p}(G) are all the maps induced by inner automorphisms of G.

Definition 2.3.2. Let G be a finite group and A_{p}(G) be its Quillen category. A subcategory Λ of A_{p}(G) is said to be proper Quillen subcategory for G if

0. It is a full subcategory.

1. If E_{1} → E ← E_{2} is a diagram in A_{p}(G) with E_{i} ∈ Λ and E is generated by the images of E_{1} and E_{2}, then Λ contains an object isomorphic to E.

2. If E ∈ Λ and τ is a Sylow p-subgroup of Mor(E; E) then E^{τ} ∈ Λ.

3. There exists E ∈ Λ such that C_{G}(E) contains a Sylow p-subgroup of G.

For any group G, let EG denote some contractible free G-space.

Theorem 2.3.1 (Jackowsky, McClure). Let G be a finite group and p a fixed prime number. Let Λ be a proper Quillen subcategory for G. Then the natural map

hocolim_{Λ^{op}}EG ×_{G} G/C_{G}(−) ⟶ BG

induces a mod-p cohomology isomorphism.

2.4. The Neisendorfer Fibre Square Lemma

Lemma 2.4.1 below is a particular case of a theorem due to J. Neisendorfer [52, prop. 7.2]. We state it for future reference.

Lemma 2.4.1. Let

[figure: E_{1}; E_{2}; E_{3}; E_{4}; B_{1}; B_{2}; B_{3}; B_{4}] (2.4.1)

be a commutative diagram of maps and spaces, where the top and bottom faces are homotopy pull-back diagrams. Let F_{i} be the homotopy fibre of E_{i} ⟶ B_{i}. Then the diagram of fibres:

[figure: F_{1}; F_{2}; F_{3}; F_{4}]

is a homotopy pull-back square.

CHAPTER 3 Resolutions by Fibrations

In this chapter we recall the notion of resolutions by fibrations over an arbitrary family of spaces and discuss some related facts. Here by a space we mean a connected compactly generated Hausdorf topological space. The idea of resolving a space by fibrations is elementary, but for various applications it may become useful. Most of the results in this chapter are either well-known or straightforward. However we spell them out, as the purpose this chapter is to set up a systematic background for the rest of this part of the paper.

3.1. Definition and Basic Examples

Definition 3.1.1. Let X be a space. Let C be an arbitrary family of connected spaces. We say that X is resolvable by fibrations over C if there exists a tower of principal fibrations

⋯ ⟶ X_{n} ⟶ X_{n−1} ⟶ ⋯ ⟶ X_{2} ⟶ X_{1} = X,

such that

1. For each i ≥ 1 there exists B_{i} ∈ C such that the fibration, X_{i+1} ⟶ X_{i}, is induced from the path loop fibration over B_{i} via a map ρ_{i} : X_{i} ⟶ B_{i}.

2. lim_{\overleftarrow{i}} X_{i} is weakly contractible.

In particular we say that X is resolvable by fibrations over C of length ≤ n if there exists a tower, as above, such that X_{n} is homotopy equivalent to an object of C.

We will sometimes abbreviate the notion “resolvable by fibrations over C” by “C-resolvable” and “C-resolvable of finite length” by “finitely C-resolvable”. If X is a C-resolvable space then any associated resolution will be called a C-resolution for X. If X is finitely C-resolvable then we denote by l_{C}(X) the minimal length of a C-resolution for X.

Example 3.1.1. Let K be the family of all Eilenberg-MacLane spaces. Then any space X is resolvable by fibrations over K, with X_{n} = X⟨n − 1⟩, the (n − 1)- connected cover of X. Furthermore, if P_{n}X is the n-th Postnikov stage for X then P_{n}X is K-resolvable of length ≤ n.

Example 3.1.2. Let F be the family of finite CW-complexes. Let B be the classifying space of a non-trivial finite group. Then B is not F-resolvable. This is an immediate consequence of Miller’s theorem on the Sullivan conjecture [48], and proposition 3.1.1 below.

Proposition 3.1.1. Let X be a non-contractible CW-complex and C, a family of spaces. Suppose that for any Y ∈ C, the mapping space Map_{∗}(X; Y) is weakly contractible. Then X does not admit a C-resolution.

Proof. Let

⋯ ⟶ X_{n} ⟶ X_{n−1} ⟶ ⋯ ⟶ X_{2} ⟶ X_{1} = X

be a C-resolution for X. Since for every B ∈ C, Map_{∗}(X; B) is weakly contractible, it follows that the identity map on X lifts through the tower. In particular we get a map

X ⟶ lim_{\overleftarrow{i}} X_{i}

inducing a monomorphism on homotopy groups. Since X is non-contractible by assumption, it follows that lim_{\overleftarrow{i}} X_{i} is not weakly contractible.

If C has some special feature, then the existence of a C-resolution for a space X might imply that X has the same property. An example for that is given in the following

Proposition 3.1.2. Let C be a family of spaces whose objects have homotopy exponents. Let X be a finitely C-resolvable space. Then X has a homotopy exponent.

Proof. Let

X_{n} ⟶ X_{n−1} ⟶ ⋯ ⟶ X_{2} ⟶ X_{1} = X

be a finite C-resolution for X, such that X_{i+1} ⟶ X_{i} is classified by a map X_{i} ⟶ B_{i}. Suppose that B_{i} has a homotopy exponent k_{i}. In particular X_{n} ≃ B_{n}. By the long exact homotopy sequence for the fibration

X_{n} ⟶ X_{n−1} ⟶ B_{n−1}

it follows that X_{n−1} has a homotopy exponent bounded above by k_{n} · k_{n−1}. By induction, X has a homotopy exponent bounded above by k = ∏_{i=1}^{n} k_{i}.

3.2. A Fibration Lemma

Suppose C is an arbitrary family of spaces. The problem of either deciding which spaces are C-resolvable or of possibly constructing a resolution for a given space X, is by no means obvious. Thus the following proposition might become useful in giving a positive answer for a particular case. The proposition can be stated in greater generality. However, we restrict attention to finite resolutions.

Lemma 3.2.1. Let C be a family of spaces. Let F ⟶ E ⟶^{j_{1}} X be a fibration. If F and X are finitely C-resolvable, with l_{C}(F) = n and l_{C}(X) = k, then E is finitely C-resolvable with l_{C}(E) ≤ n + k

Proof. Let

X_{k} ⟶^{q_{k−1}} X_{k−1} ⟶^{q_{k−2}} ⋯ ⟶^{q_{2}} X_{2} ⟶^{q_{1}} X_{1} = X

and

F_{n} ⟶^{l_{n−1}} F_{n−1} ⟶^{l_{n−2}} ⋯ ⟶^{l_{2}} F_{2} ⟶^{l_{1}} F_{1} = F

be finite C-resolutions for X and F respectively, such that q_{i} is classified by a map ρ′_{i} : X_{i} ⟶ B_{i} and l_{i} is classified by a map ρ′′_{i} : F_{i} ⟶ C_{i}.

Let E_{2} ⟶^{p_{1}} E_{1} be the fibration defined by the pull-back square

[figure: E_{2}; p_{1}; X_{2}; q_{1}; j_{2}; E_{1}; X_{1}; j_{1}] (3.2.1)

One routinely verifies that E_{2} ⟶^{p_{1}} E_{1} is also the principal fibration, induced from the path loop fibration over B_{1} via the composite:

E_{1} ⟶^{j_{1}} X_{1} ⟶^{ρ′_{1}} B_{1}.

Furthermore, since (3.2.1) is a pull-back diagram, E_{2} ⟶^{j_{2}} X_{2} is a fibration with fibre F.

Inductively, let E_{i+1} ⟶^{p_{i}} E_{i} be the fibration defined by the pull-back square [figure: E_{i+1}; p_{i}; X_{i+1}; q_{i}; j_{i+1}; E_{i}; X_{i}; j_{i}] (3.2.2)

As for i = 1, E_{i+1} ⟶^{p_{i}} E_{i} is the principal fibration induced from the path loop fibration over B_{i} via the composite:

E_{i} ⟶^{j_{i}} X_{i} ⟶^{ρ′_{i}} B_{i}

and since (3.2.2) is a pull-back square, E_{i+1} ⟶^{j_{i+1}} X_{i+1} is a fibration with fibre F.

By hypothesis, the k-th classifying map X_{k} ⟶^{ρ′_{k}} B_{k} is a homotopy equivalence. Thus X_{k+1} is contractible and in the k-th iteration of the process we get a pullback diagram

[figure: E_{k+1}; p_{k}; X_{k+1} ≃ ∗; q_{k}; j_{k+1}; E_{k}; X_{k}; j_{k}]

Thus the fibre inclusion F = F_{1} ⟶ E_{k+1} is a homotopy equivalence. Let E_{k+1} ⟶^{e_{1}} F_{1} be some homotopy inverse. Define E_{k+2} ⟶^{e_{k+1}} E_{k+1} to be the fibration defined by the pull-back square:

[figure: E_{k+2}; p_{k+1}; F_{2}; l_{1}; e_{2}; E_{k+1}; F_{1}; e_{1}] (3.2.3)

Since e_{1} is an equivalence and (3.2.3) is a pull-back square, it follows that E_{k+2} ⟶^{e_{2}} F_{2} is a homotopy equivalence. As before, one easily verifies that E_{k+2} ⟶^{p_{k+1}} E_{k+1} is the principal fibration induced from the path loop fibration over C_{1} via the composite:

E_{k+1} ⟶^{e_{1}} F_{1} ⟶^{ρ′′_{1}} C_{1}.

Iteration of this step m times now completes the proof.

3.3. THE MOD-p COHEN CONJECTURE 61

3.3. The mod-p Cohen Conjecture

Definition 3.3.1. A family C of spaces is said to be Ω-closed if for any simply-connected space X ∈ C, ΩX ∈ C. If C is an arbitrary family of spaces, then the Ω-closure of C is defined to be the family Ω̄C of all the spaces of the form ΩX with X ∈ C simply-connected.

Let S′ denote the family of all spheres together with the one point space. Let S denote the Ω-closure of S′. In [17], F. Cohen stated a “rash conjecture”

Conjecture 3.3.1 (Cohen). Let π be a finite super-perfect group. Then ΩBπ^{∧} is finitely S-resolvable.

As we have already observed, the study of homotopy properties of ΩBπ^{∧} is a p-local problem in nature. Thus one might like to restate conjecture 3.3.1 in terms of p-perfect groups and replace the functor (−)^{∧} by (−)_{p}^{∧}. However, we have already remarked in the introduction to this part that a counter example to the mod-p Cohen’s conjecture was recently discovered. One might wonder whether the conjecture is true if further restrictions are imposed on the group π. For example it would be rather interesting to know that the conjecture is true for all finite simple groups. Through the rest of this part of the paper, our major task will be to obtain finite S-resolutions for spaces of type ΩBπ_{p}^{∧}, where π is a finite p-perfect group. A more detailed discussion of Cohen’s conjecture is given in chapter 7 below.

62 3. RESOLUTIONS BY FIBRATIONS

CHAPTER 4 Sporadic Examples

In this chapter we consider spaces of the form ΩBπ_{p}^{∧} from the point of view of the Sylow p-subgroup of π. More precisely, we have two basic assumptions:

1. π is p-perfect and

2. The Sylow p-subgroup of π is “well understood”.

Given this data, we study various properties of ΩBπ_{p}^{∧} and in particular the existence of an S-resolution.

A remark is due here. Our treatment depends heavily on the highly nontrivial theory of [30]. It is quite likely that the problems we solve using this theory can more easily be sorted out by means of group theoretic techniques. However, as the reader may observe, we are deducing group theoretic facts on a given group π with minimum assumptions and using a theory of a topological nature. As group theory is by no means a self contained and “closed” subject, a justification of our rather lengthy treatment is in that the same tools may apply to further examples where purely group theoretic tools have so far failed.

The examples below are constructed at the prime 2, but there is no important reason to restrict to this prime. Throughout this chapter, we assume that the reader is familiar with [30] and refer back to section 2.3 for a brief account of the main result of this paper. We shall write H^{∗}(−) for H^{∗}(−; 𝔽_{2}).

4.1. Groups with a Dihedral Sylow 2-Subgroup

Recall that the dihedral group of order 2^{n} is given by

D_{2^{n}} = ⟨x, y|x^{2^{n−1}} = y^{2} = 1; yxy^{−1} = x^{−1}⟩.

We shall also use the quaternion group of order 2^{n+1}. Recall that

Q_{2^{n+1}} = ⟨α, β|α^{2^{n−1}} = β^{2}; α^{2^{n}} = 1; βαβ^{−1} = α^{−1}⟩.

We restate theorem 1.1.1 as

Theorem 4.1.1. Let π be a finite 2-perfect group containing the dihedral group D_{2^{n}} as a Sylow 2-subgroup. Then the 1-connected cover ΩBπ_{p}^{∧}⟨1⟩ is homotopy equivalent to S^{3}{2^{n+1}} and thus is S-resolvable of length 2.

For the proof we need to recall from [17]

Proposition 4.1.1 (Cohen). Let π be a finite 2- perfect group containing a quaternion group Q_{2^{n+1}} as a Sylow 2-subgroup. Then ΩBπ_{2}^{∧} ≃ S^{3}{2^{n+1}}.

In view of proposition 2.1.1, theorem 4.1.1 now follows immediately from the following purely group theoretic fact

Proposition 4.1.2. Let π be a finite 2-perfect group containing D_{2^{n}} as a Sylow 2-subgroup. Let π̃ be the 2-universal central extension of π. Then the Sylow 2-subgroup of π̃ is isomorphic to Q_{2^{n+1}}.

The proof of proposition 4.1.2 will occupy the rest of this section.

There is a central extension:

0 ⟶ ℤ/2ℤ ⟶ Q_{2^{n+1}} ⟶^{p} D_{2^{n}} ⟶ 1 (4.1.1)

where p(α) = x and p(β) = y. The cohomology of D_{2^{n}} is well known [26].

H^{∗}(BD_{2^{n}}) = \{ \begin{matrix}P [x_{1}, y_{1}, w_{2}]/(x^{2} + xy) & n > 2 \\ P [x_{1}, y_{1}] & n = 2 \end{matrix}

where x_{1} and y_{1} are the duals of the respective homology classes. The class w_{2} is the second Stiefel-Whitney class of the representation ρ : D_{2^{n}} ⟶ O(2) [26, VI.3], given in section 5.1 of part 1.

Let σ ∈ H^{2}(BD_{2^{n}}) be the characteristic class for (4.1.1) above. By [50], σ = w + y^{2} for n > 2 and σ = x^{2} + xy + y^{2} if n = 2. In particular, let D_{2^{n−1}} < D_{2^{n}} , n ≥ 3, be the dihedral subgroup generated by x^{2} and y. Then Res(D_{2^{n}}, D_{2^{n−1}}) preserves σ.

Let π be a finite 2-perfect group containing D_{4} as a Sylow 2-subgroup. By example 2.2.1, Bπ_{2}^{∧} ≃ (BA_{5})_{2}^{∧}. The universal central extension of A_{5} is given by the group SL_{2}(𝔽_{5}) and thus there is a principal fibration

Bℤ/2ℤ ⟶ BSL_{2}(𝔽_{5}) ⟶ BA_{5}. (4.1.2)

Let τ : Bπ_{2}^{∧} ⟶ (BA_{5})_{2}^{∧} be any equivalence. Let π̃ be the 2-universal central extension of π. Since BSL_{2}(𝔽_{5})_{2}^{∧} is the 2-connected cover of (BA_{5})_{2}^{∧} and Bπ̃_{2}^{∧} is the 2-connected cover of Bπ_{2}^{∧} it follows that τ induces an equivalence τ̃ : Bπ̃_{2}^{∧} ≃ BSL_{2}(𝔽_{5})_{2}^{∧}. In particular it follows from [51] that π̃ and SL_{2}(𝔽_{5}) have isomorphic Sylow 2-subgroup. Since the Sylow 2-subgroup of SL_{2}(𝔽_{5}) is isomorphic to Q_{8}, proposition 4.1.2 is true for n = 2.

Notice in particular that, in view of [50], this result implies that if π is a finite 2-perfect group containing D_{4} as a Sylow 2-subgroup, then the single nonvanishing class v_{2} ∈ H^{2}(Bπ) restricts to σ ∈ H^{2}(BD_{4}), the characteristic class for (4.1.1).

Before proceeding, we describe our strategy. First we show that if π is any finite 2-perfect group whose Sylow 2-subgroup is isomorphic to D_{2^{n}}, then there exists a 2-perfect subgroup π′ < π containing D_{4} as a Sylow 2-subgroup. We use this fact to deduce that the characteristic class for (4.1.1) above is in the image of Res(π, D_{2^{n}}). Thus the corresponding classifying map σ : BD_{2^{n}} ⟶ K(ℤ/2ℤ, 2) extends to give a commutative diagram:

[figure: BD_{2^{n}}; K(ℤ/2ℤ, 2); =; σ; Bπ; K(ℤ/2ℤ, 2); v_{2}]

Finally we show that the homotopy fibre of v_{2} is the classifying space for π̃, the 2-universal central extension of π. In particular this implies that the Sylow 2subgroup of π̃ is isomorphic to Q_{2^{n+1}} and thus completes the proof of proposition 4.1.2 and theorem 4.1.1.

Lemma 4.1.1. Let π be a finite group containing D_{2^{n}} as a Sylow 2-subgroup. If π contains a central element of order 2 then the mod-2 cohomology restriction map Res(π, D_{2^{n}}) is an isomorphism.

Proof. For n = 2, the Weyl group W_{π}(D_{4}) is either trivial or isomorphic to ℤ/3ℤ. Thus in this case the result follows. Assume n ≥ 3. By [41], H^{∗}(Bπ) has exactly three possible structures as a module over the Steenrod algebra. These are uniquely determined by dim_{𝔽_{2}}H^{1}(Bπ), which can be 0, 1 or 2. We first show that if π contains a central element of order 2 then it cannot be 2-perfect. Suppose π is a 2-perfect group. By [50] and [41], H^{∗}(Bπ) can have either one of the following algebra structures:

H^{∗}(π) ≅ \{ \begin{matrix}P [v_{2}, v_{3}, v̄_{3}]/(v̄_{3}^{2} + v_{3}v̄_{3}) & or \\ P [v_{2}, v_{3}, v̄_{3}]/(v̄_{3}^{2} + v_{3}^{2} + v_{3}v̄_{3} + v_{2}^{3}). \end{matrix}

In particular, Sq^{1}(v_{2}) = v_{3}. Let a ∈ π be a central element of order 2. Let E = ⟨a⟩, and π′ = π/E. Since π is 2-perfect, so is π′. Furthermore, the element a is contained in a Sylow 2-subgroup of π and is central there. Since the quotient of D_{2^{n}} by its center is isomorphic to D_{2^{n−1}}, it follows that π′ contains D_{2^{n−1}} as a Sylow 2-subgroup. Now H^{2}(Bπ′) ≅ ⟨v_{2}⟩ℤ/2ℤ. Thus the only non-trivial central extension of π′ by ℤ/2ℤ is the 2-universal central extension. But

0 ⟶ E ⟶ π ⟶ π′ ⟶ 1

is a central extension of π′ by ℤ/2ℤ. Thus π is isomorphic to the 2-universal central extension of π′. However, H^{2}(Bπ) ≠ 0. By proposition 2.1.1, we get a contradiction.

Suppose now that dim_{𝔽_{2}}H^{1}(Bπ) = 1 and assume that there exists a central subgroup E < π, of order 2, as above. Let g : Bπ ⟶ K(ℤ/2ℤ, 1) correspond to the generator of H^{1}(Bπ). Let B̃ be the homotopy fibre of g. By the exact homotopy sequence, it follows that B̃ is, up to homotopy, the classifying space of a subgroup π̃ < π. Furthermore, since ℤ/2ℤ ≅ π_{1}K(ℤ/2ℤ, 1) acts nilpotently on H^{∗}(B̃), it follows that

B̃_{2}^{∧} ⟶ Bπ_{2}^{∧} ⟶^{g} K(ℤ/2ℤ, 1)

is a fibration. In particular, since Sq^{1}g ≠ 0 [50], it follows that g induces an isomorphism on fundamental groups. Thus B̃_{2}^{∧} is the 1-connected cover of Bπ_{2}^{∧} and so π̃ is 2-perfect. Notice that the order of the Sylow 2-subgroup of π̃ is 2^{n−1}. Now, by [58], the Sylow 2 -subgroup of π̃ is cyclic if and only if π̃ has periodic mod-2 cohomology of period 2. In particular, in this case π̃ cannot be 2-perfect. Consequently, the Sylow 2-subgroup of π̃ must be a non-cyclic subgroup of D_{2^{n}}. But the only non-cyclic subgroup of the dihedral group D_{2^{n}}, whose order is 2^{n−1}, is the subgroup generated by x^{2} and y, which is isomorphic to D_{2^{n−1}}.

Now, E is a central 2-subgroup of π. Hence it is contained in the center of a Sylow 2-subgroup and we may assume that E = Z(D_{2^{n}}) = ⟨x^{2^{n−2}}⟩. On the other hand, every dihedral subgroup of D_{2^{n}} contains Z(D_{2^{n}}). Thus π̃ > E as a central subgroup. This contradicts our first assertion and thus completes the proof.

Let T, K < D_{2^{n}} be the subgroups generated by {x^{2^{n−2}}, y} and {x^{2^{n−2}}, xy} respectively.

Lemma 4.1.2. Let π be a finite group containing D_{2^{n}} as a Sylow 2-subgroup. Then W_{π}T and W_{π}K are isomorphic to non-trivial subgroups of Σ_{3}.

Proof. We prove the statement for T. The proof for K is essentially the same. Since T is a maximal elementary abelian subgroup of D_{2^{n}}, it follows that the Sylow 2-subgroup of C_{π}T is T itself. Next, T is contained as a normal subgroup in the subgroup D_{8} of D_{2^{n}} generated by x^{2^{n−3}} and y. Thus D_{8} < N_{π}T. It follows that W_{π}T = N_{π}T/C_{π}T contains an element of order 2. Furthermore, W_{π}T < Aut(T) ≅ Σ_{3}. This completes the proof.

In the proof of the following proposition we assume that the reader is familiar with [30].

Proposition 4.1.3. Let π be a finite 2-perfect group containing D_{2^{n}}, n ≥ 3 as a Sylow 2-subgroup. Let T, K < D_{2^{n}} be as in lemma 4.1.2. Then W_{π}T and W_{π}K are isomorphic to Σ_{3}.

Proof. By lemma 4.1.2, it suffices to show that W_{π}T and W_{π}K cannot be isomorphic to ℤ/2ℤ. We carry out the proof for T. The result for K follows in essentially the same way.

Let E < D_{2^{n}} be the subgroup generated by x^{2^{n−2}}. Then E = Z(D_{2^{n}}) and so C_{π}(E) contains D_{2^{n}} as a Sylow 2-subgroup. Suppose that W_{π}T ≅ ℤ/2ℤ. We derive a contradiction by using [30] to show that in this case π cannot be 2-perfect. Consider the full subcategory Γ of the Quillen category A_{2}(π), whose objects are T and E. Claim: The morphisms in Γ are given as follows:

Mor(T; T) ≅ W_{π}T; Mor(E; E) = {id}; Mor(E; T) = {E ↪^{i} T }

The only point we need to justify is that Mor(E; T) consists of the natural inclusion i : E ⟶ T only. Suppose that there is another morphism ϕ ∈ Mor(E; T) different from i. Let α denote the non-trivial morphism in Mor(T; T). Notice that α can be represented by the element x^{2^{n−3}} ∈ D_{2^{n}} and thus it permutes the elements x^{2^{n−2}}y and y in T and fixes x^{2^{n−2}}. Thus the morphisms i, ϕ and αϕ are all three possible embedings of E in T. We will show that this implies the existence of a non-trivial element β ∈ Mor(T; T) different from α, thus contradicting the hypothesis.

Let ϕ ∈ Mor(E; T) be given by ϕ(x^{2^{n−2}}) = y. Let g ∈ π be an element representing ϕ. Thus gx^{2^{n−2}}g^{−1} = y

Let E′ be the subgroup of π generated by y. Consider the centralizer C_{π}(E′) = gC_{π}(E)g^{−1}. Let γ = gxg^{−1} and δ = gyg^{−1}. Then γ and δ generate a Sylow 2-subgroup B of C_{π}(E′). Notice that γ^{2^{n−2}} = gx^{2^{n−2}}g^{−1} = y. Since x^{2^{n−2}} commutes with y, it is a member of C_{π}(E′). Thus there exist some h ∈ C_{π}(E′) such that hx^{2^{n−2}}h^{−1} ∈ B.

Write hx^{2^{n−2}}h^{−1} = γ^{k}δ^{ϵ} for some 0 ≤ k ≤ 2^{n−1} − 1 and 0 ≤ ϵ ≤ 1. If k = 0 then hx^{2^{n−2}}h^{−1} = δ. Thus we get g^{−1}hx^{2^{n−2}}h^{−1}g = g^{−1}δg = y and g^{−1}hyh^{−1}g = g^{−1}yg = x^{2^{n−2}}. Restricting conjugation by g^{−1}h to T we obtain an automorphism which permutes y and x^{2^{n−2}} and fixes x^{2^{n−2}}y. By hypothesis such a morphism does not exist.

Suppose ϵ = 0. Then hx^{2^{n−2}}h^{−1} = γ^{k}. Thus γ^{2k} = 1, so k = 2^{n−2} and hence hx^{2^{n−2}}h^{−1} = γ^{2^{n−2}} = y. But h ∈ C_{π}(E′) and E′ is generated by y. Thus it follows that x^{2^{n−2}} = y which is impossible. Hence we must have hx^{2^{n−2}}h^{−1} = γ^{k}δ for some k > 0. Conjugating by g^{−1}, we get g^{−1}hx^{2^{n−2}}h^{−1}g = g^{−1}γ^{k}δg = x^{k}y.

Let ψ denote the inner automorphism given by conjugation by g^{−1}h. Write β = ψ^{−1}αψ. One routinely verifies that the restriction of β to T defines an automorphism which permutes x^{2^{n−2}} and x^{2^{n−2}}y and fixes y. Again, by hypothesis, such an automorphism does not exist in Mor(T; T). This completes the proof of the claim.

One now easily checks definition 2.3.2 to verify that Γ is a proper Quillen subcategory of A_{2}(π). Thus by theorem 2.3.1, the natural map:

H^{∗}(Bπ) ⟶ lim_{\overleftarrow{Γ}} H^{∗}(BC_{π}(−))

is an isomorphism.

By hypothesis Mor(T; T) ≅ ℤ/2ℤ. Let α ∈ Mor(T; T) and i ∈ Mor(E; T) be as above. Write

ᾱ : C_{π}T ⟶ C_{π}T and ī: C_{π}T ⟶ C_{π}E

for a choice of induced homomorphisms. Notice that ᾱ and ī are determined up to conjugation by an element of the target. Thus their cohomological effect is uniquely determined. Now, the inverse limit can be computed as the kernel of the map

φ : H^{∗}(BC_{π}E) × H^{∗}(BC_{π}T) ⟶ H^{∗}(BC_{π}T)^{×2}

given by φ(a, b) = (b − ī^{∗}a, b − ᾱ^{∗}b).

By lemma 4.1.1, H^{∗}(BC_{π}E) ≅ H^{∗}(BD_{2^{n}}). By the proof of lemma 4.1.2, T < C_{π}T as a central Sylow 2-subgroup. Thus H^{∗}(BC_{π}T) ≅ H^{∗}(BT). Therefore ī^{∗} is not injective in dimension 1 [26]. Let z ∈ H^{1}(BC_{π}E) be a non-zero element in Ker(ī^{∗}). Then φ(z, 0) = 0. Thus (z, 0) represents a non-zero element of H^{1}(Bπ), so π cannot be 2-perfect. This completes the proof.

Corollary 4.1.1. Let π be a finite 2-perfect group containing D_{2^{n}} as a Sylow 2-subgroup. Then there exist 2-perfect subgroups G_{1}, G_{2} < π containing T and K respectively as Sylow 2-subgroups.

Proof. As before, we prove the corollary for T and remark that the result for K follows by analogy.

By proposition 4.1.3 there is a short exact sequence of groups

1 ⟶ C_{π}T ⟶ N_{π}T ⟶ Σ_{3} ⟶ 1.

Define G_{1} to be the pull-back in the diagram:

[figure: G_{1}; A_{3}; p′; N_{π}T; Σ_{3}; p]

G_{1} contains T as a normal Sylow 2-subgroup and the Weyl group W_{G_{1}}(T) is isomorphic to A_{3}. Thus by corollary 2.2.1,

(BG_{1})_{2}^{∧} ≃ B(T ⋊ W_{G_{1}}(T))_{2}^{∧}.

By example 2.2.1, (BG_{1})_{2}^{∧} ≃ (BA_{5})_{2}^{∧}. In particular G_{1} is 2-perfect. This completes the proof.

Proposition 4.1.4. Let π be a finite 2-perfect group containing D_{2^{n}}, n ≥ 3 as a Sylow 2-subgroup. Then, with the notations above, the mod-2 cohomology restriction map sends v_{2} to σ, the characteristic class for the extension ( 4.1.1).

Proof. Let G_{1}, G_{2} < π be the 2-perfect subgroups of π containing T and K as Sylow 2-subgroups respectively. There are commutative diagrams given as follows.

[figure: T; G_{1}; K; G_{2}; D_{2^{n}}; π; D_{2^{n}}; π]

The mod-2 cohomology restriction maps into the cohomology of the respective Sylow 2-subgroups are, of course, injective. By [50], the restriction maps Res(D_{2^{n}}, T) and Res(D_{2^{n}}, K) are given by

x y  w_{2}
T 0 y x^{2} + xy
K y y x^{2} + xy

Since (BG_{1})_{2}^{∧} ≃ (BG_{2})_{2}^{∧} ≃ (BA_{5})_{2}^{∧}, it follows that

Res(G_{1}, T)(v_{2}) = σ and Res(G_{2}, K)(v_{2}) = σ.

We claim that Res(π, G_{1})(v_{2}) = v_{2} and Res(π, G_{2})(v_{2}) = v_{2}.

Suppose Res(π, G_{1})(v_{2}) = 0. Then

Res(π, D_{2^{n}})(v_{2}) ∈ Ker(Res(D_{2^{n}}, T)) = ⟨x^{2}⟩.

But if Res(π, D_{2^{n}})(v_{2}) = x^{2}, we have

Res(π, D_{2^{n}})(v_{3}) = Sq^{1}Res(π, D_{2^{n}})(v_{2}) = Sq^{1}x^{2} = 0.

This contradicts injectivity of Res(π, D_{2^{n}}).

Similarly, suppose that Res(π, G_{2})(v_{2}) = 0. Then

Res(π, D_{2^{n}})(v_{2}) ∈ Ker(Res(D_{2^{n}}, K)) = ⟨x^{2} + y^{2}⟩.

The same argument leads again to a contradiction.

Now, since Res(π, G_{1})(v_{2}) = v_{2}, we have

Res(π, D_{2^{n}})(v_{2}) = σ + ax^{2}, a ∈ 𝔽_{2}.

Since Res(π, G_{2})(v_{2}) = v_{2},

Res(π, D_{2^{n}})(v_{2}) = σ + b(x^{2} + y^{2}), b ∈ 𝔽_{2}.

Thus a = b = 0 and we have Res(π, D_{2^{n}})(v_{2}) = σ. This completes the proof.

We are now able to prove proposition 4.1.2. Let π be a finite 2-perfect group containing D_{2^{n}}, n ≥ 3, as a Sylow 2-subgroup. Since Res(π, D_{2^{n}})(v_{2}) = σ, we have a commutative diagram:

[figure: BD_{2^{n}}; K(ℤ/2ℤ, 2); ≃; σ; Bπ; K(ℤ/2ℤ, 2); v_{2}]

Let B̃ be the homotopy fibre of v_{2}. The non vanishing part of the long exact homotopy sequence for

B̃ ⟶ Bπ ⟶ K(ℤ/2ℤ, 2)

has the form of a non-split central extension:

0 ⟶ ℤ/2ℤ ⟶ π_{1}B̃ ⟶ π ⟶ 1 (4.1.3)

Thus π̃ = π_{1}B̃ is the 2-universal central extension of π. Finally, the Sylow 2subgroup of π̃ is the central extension of D_{2^{n}} by ℤ/2ℤ corresponding to σ. Thus π̃ contains Q_{2^{n+1}} as a Sylow 2-subgroup. This completes the proof of proposition 4.1.2 and thus of theorem 1.1.1.

We end this section in computing the homology of ΩBπ_{2}^{∧}, where π is a finite 2-perfect group containing D_{2^{n}} as a Sylow 2-subgroup.

Proposition 4.1.5. Let π be a finite 2-perfect group containing D_{2^{n}} as a Sylow 2-subgroup. Then H_{∗}(ΩBπ_{2}^{∧}) ≅ E[y_{1}, y_{2}] ⊗ P [α_{2}] as an algebra, where the degrees of the generators are given by their subscripts. The Hopf algebra structure is determined by the requirement that α_{2} is primitive and the reduced diagonal of y_{2} is given by Δ̃(y_{2}) = y_{1} ⊗ y_{1}. The primary Bockstein operators are given by Sq_{∗}^{1}(y_{2}) = y_{1} and Sq_{∗}^{1}(α_{2}) = 0.

Proof. By proposition 4.1.2, the 2-universal central extension π̃ of π contains the quaternion group Q_{2^{n+1}} as a Sylow 2-subgroup. Thus we obtain a principal fibration

Bℤ/2ℤ ⟶^{j} Bπ̃_{2}^{∧} ⟶^{p} Bπ_{2}^{∧}. (4.1.4)

Write H^{∗}(Bπ̃) ≅ P [p_{4}] ⊗ E[u_{3}] and H^{∗}(Bℤ/2ℤ) ≅ P [z_{1}]. Then j^{∗}(u_{3}) = 0 and j^{∗}(p_{4}) = z_{1}^{4}, [25]. Next, (4.1.4) gives rise to a principal fibration of H-spaces and H-maps

ΩBπ̃_{2}^{∧} ⟶^{Ωp} ΩBπ_{2}^{∧} ⟶^{δ} Bℤ/2ℤ. (4.1.5)

By proposition 4.1.1, ΩBπ̃_{2}^{∧} ≃ S^{3}{2^{n+1}}. Thus H^{∗}(ΩBπ̃_{2}^{∧}) ≅ Γ[u_{2}] ⊗ E[y_{3}] and in the mod-2 cohomology Serre spectral sequence we have d_{3}(u_{2}) = 0 and d_{4}(y_{3}) = z_{1}^{4}. Furthermore, d_{4}(z_{1}^{k}y_{3}) = z_{1}^{k}d_{4}(y_{3}) = z_{1}^{k+1}. Consider the mod-2 homology Serre spectral sequence for (4.1.5). We have H_{∗}(ΩBπ̃) ≅ P [μ_{2}]⊗E[ν_{3}] and H_{∗}(Bℤ/2ℤ) ≅ Γ[x_{1}]. Thus

E^{2} ≅ Γ[x_{1}] ⊗ (P [μ_{2}] ⊗ E[ν_{3}]),

and by duality

d_{2}(γ_{2}(x_{1})) = d_{3}(γ_{3}(x_{1})) = 0; d_{4}(γ_{4}(x_{1})) = ν_{3}; and d_{4}(γ_{k+4}(x_{1})) = γ_{k}(x_{1})ν_{3}.

Since (4.1.5) is a multiplicative fibration, this determines the spectral sequence. Indeed we have d_{4}(μ_{2}^{l}γ_{k+4}(x_{1})) = μ_{2}^{l}d_{4}(γ_{k+4}(x_{1})) = μ_{2}^{l}ν_{3}γ_{k}(x_{1}). Thus

E^{5} ≅ E^{∞} ≅ E[y_{1}, y_{2}] ⊗ P [α_{2}]

and, additively we may write

H_{∗}(ΩBπ_{2}^{∧}) ≅ E[y_{1}, y_{2}] ⊗ P [α_{2}] (4.1.6)

where the classes y_{i}, i = 1, 2, are determined by the requirement that δ_{∗}(y_{i}) = γ_{i}(x_{1}) ∈ H_{∗}(Bℤ/2ℤ) and α_{2} is determined to be the image of the element μ_{2} ∈ H_{2}(ΩBπ̃_{2}^{∧}) under Ωp_{∗}. It follows immediately that (4.1.6) is an isomorphism of P [α_{2}]-modules. Since Sq_{∗}^{1}(μ_{2}) = 0, and Sq_{∗}^{1}(γ_{2}(x_{1})) = γ_{1}(x_{1}), it follows that Sq_{∗}^{1}(y_{2}) = y_{1} and Sq_{∗}^{1}(α_{2}) = 0.

We proceed in computing the mod-2 cohomology Serre spectral sequence for the path-loop fibration over Bπ_{2}^{∧}. By [50] we can write

H^{∗}(Bπ_{2}^{∧}) ≅ P [v_{2}, v_{3}, v̄_{3}]/(ρ)

where the relation ρ is given by either v̄_{3}^{2} + v_{3}v̄_{3} = 0 or v̄_{3}^{2} + v_{3}v̄_{3} + v_{3}^{2} + v_{2}^{3} = 0. Furthermore Sq^{1}(v_{2}) = v_{3}, Sq^{2}(v_{3}) = v_{2}v_{3} and Sq^{2}(v̄_{3}) = either v_{2}v̄_{3} or v_{2}v̄_{3} + v_{2}v_{3}, depending on the relation ρ. Additively there is an isomorphism

H^{∗}(ΩBπ_{2}^{∧}) ≅ E[ŷ_{1}, ŷ_{2}] ⊗ Γ[α̂_{2}].

Thus the Serre spectral sequence takes the form

E_{2} ≅ P [v_{2}, v_{3}, v̄_{3}]/(ρ) ⊗ E[ŷ_{1}, ŷ_{2}] ⊗ Γ[α̂_{2}]

as a differential (P [v_{2}, v_{3}, v̄_{3}]/(ρ))-module. We have d_{2}(ŷ_{1}) = v_{2} for dimension reasons. Thus d_{3}(ŷ_{2}) = Sq^{1}(d_{2}(ŷ_{1})) = Sq^{1}(v_{2}) = v_{3}. Also if α̂_{2} was not transgressive then v̄_{3} would be an infinite cycle. Thus d_{3}(α̂_{2}) = v̄_{3}. Consequently we have

E_{3} ≅ P [v_{3}, v̄_{3}]/ρ′ ⊗ (E[ŷ_{2}] ⊗ Γ[α̂_{2}]),

where ρ′ is equal to either v̄_{3} + v_{3}v̄_{3} or v̄_{3} + v_{3}v̄_{3} + v_{3}^{2}, depending on ρ. Notice that P [v_{3}, v̄_{3}]/ρ′ ≅ P [v_{3}]⊗E[v̄_{3}] as a P [v_{3}]-module. Thus by a direct calculation, which we omit, it follows that E_{3} is acyclic and so E_{4} = E_{∞}.

A basis for H_{∗}(ΩBπ_{2}^{∧}) over 𝔽_{2}, up to dimension 4, is given as follows.

Dimension 0 1 2  3  4
1 y_{1} y_{2} y_{1}y_{2} α_{2}^{2}
α_{2} y_{1}α_{2} y_{2}α_{2}

(4.1.7)

claim: The following relations hold in H_{∗}(ΩBπ_{2}^{∧}):

y_{1}^{2} = y_{2}^{2} = 0; y_{1}y_{2} = y_{2}y_{1}; y_{1}α_{2} = α_{2}y_{1}; y_{2}α_{2} = α_{2}y_{2}.

Thus H_{∗}(ΩBπ_{2}^{∧}) is isomorphic to E[y_{1}, y_{2}] ⊗ P [α_{2}], as an algebra.

Write y_{1}^{2} = aα_{2} + by_{2}, a, b ∈ 𝔽_{2}. Since ΩBπ_{2}^{∧} ⟶^{δ} Bℤ/2ℤ is an H-map, δ_{∗}

is multiplicative. Thus δ_{∗}(y_{1}^{2}) = x_{1}^{2} = 0. On the other hand, by the calculation above, δ_{∗}(y_{1}^{2}) = bγ_{2}(x_{1}). Thus b = 0 and so y_{1}^{2} = aα_{2}. Consider the mod-2 homology Serre spectral sequence for the path loop fibration over ΩBπ_{2}^{∧}. The computation dual to the one above shows that α_{2} is in the image of the homology transgression. But if a = 1 then we have d(y_{1}v_{2}) = y_{1}d(v_{2}) = y_{1}^{2} = α_{2}. This yields a contradiction and so y_{1}^{2} = 0.

Next, let z ∈ H^{1}(Bℤ/2ℤ) denote the generator. Let ŷ_{1}, ŷ_{2} denote the duals of y_{1}, y_{2} respectively. Then δ^{∗}(z) = ŷ_{1} and δ^{∗}(z^{2}) = ŷ_{2}. Thus ŷ_{1}^{2} = ŷ_{2}. Dually the reduced homology coproduct on y_{2} is given by Δ̃(y_{2}) = y_{1} ⊗ y_{1}. Clearly y_{1} and α_{2} are primitive. Notice that a choice of y_{2} varies by α_{2}, but any choice have the same reduced coproduct.

Write y_{2}y_{1} = ay_{1}y_{2} + by_{1}α_{2}, a, b ∈ 𝔽_{2}. Applying the reduced coproduct to both sides and using the fact that y_{1}^{2} = 0 we get

y_{1} ⊗ y_{2} + y_{2} ⊗ y_{1} = a(y_{1} ⊗ y_{2} + y_{2} ⊗ y_{1}) + b(y_{1} ⊗ α_{2} + α_{2} ⊗ y_{1}).

Clearly a and b cannot be equal. If a = 0 then y_{2}y_{1} = y_{1}α_{2}. Applying the reduced coproduct again, it follows that y_{2} = α_{2} which is absurd. Thus a = 1 and b = 0, which means that y_{1} commutes with y_{2}. An identical argument applies to show that y_{1} commutes with α_{2}.

Finally, we have to show that y_{2}^{2} = 0 and that α_{2} and y_{2} commute. We conclude the proof for these two relations by considering the cohomology coproduct.

Let ŷ_{1}, ŷ_{2}, α̂_{2} denote the cohomology classes dual to y_{1}, y_{2} and α_{2} respectively. Then we have ŷ_{1}^{2} = ŷ_{2}. In the cohomology Serre spectral sequence for the pathloop fibration over ΩBπ_{2}^{∧}, E_{4} = E_{∞}. Since ŷ_{2} and α̂_{2} are transgressive, it follows that α̂_{2}^{2} = ŷ_{1}^{4} = 0. Notice that ŷ_{1}, ŷ_{2} and α̂_{2} are primitive. Thus H^{4}(ΩBπ_{2}^{∧}) is generated by ŷ_{1}^{2}α̂_{2} and a class γ_{2}(α̂_{2}) and the reduced cohomology coproducts are given by μ̃(ŷ_{1}^{2}α̂_{2}) = ŷ_{1}^{2} ⊗ α̂_{2} + α̂_{2} ⊗ ŷ_{1}^{2} and μ̃(γ_{2}(α̂_{2})) = α̂_{2} ⊗ α̂_{2}. Dualizing these equalities, we obtain the required relations. This completes the proof of our claim.

Notice that we have computed the diagonal map on the algebra generators for H_{∗}(ΩBπ_{2}^{∧}) as well. Thus the proof of the proposition is complete.

Corollary 4.1.2. Let π be a finite 2-perfect group containing D_{2^{n}} as a Sylow 2-subgroup. Then H^{∗}(ΩBπ_{2}^{∧}) ≅ P [ŷ_{1}]/(ŷ_{1}^{4} = 0) ⊗ Γ[α̂_{2}] as a Hopf algebra.

Remark 4.1.1. The computation of H_{∗}(ΩBπ_{p}^{∧}) for a finite p-perfect group π can be done also by using the Eilenberg-Moore spectral sequence. We refer the reader to the next section, where such a computation is carried out.

4.2. Groups with a Semidihedral Sylow 2-Subgroup

The semidihedral group

SD_{2^{n}} = ⟨a, b|a^{2^{n−1}} = b^{2} = 1; bab^{−1} = a^{2^{n−2}−1}⟩

is an interesting source of examples in our subject. In [40], J. Martino showed that if G is a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup then the mod-2 cohomology ring of BG is uniquely determined. Examples of 2-perfect groups whose Sylow 2-subgroup is isomorphic to SD_{2^{n}} are SL_{3}(q), q ≡ 3(4) and the Matthew group M_{11} (SD_{16}). F. Cohen [18] showed that Ω(BM_{11})_{2}^{∧} is Sresolvable of length ≤ 4. Recall that a simply-connected p-local space X is said to be of type S^{n}{p^{r}} if H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(S^{n}{p^{r}}; 𝔽_{p}) as algebras over the Steenrod algebra with the same Bockstein operators. We restate theorem 1.1.2 as

Theorem 4.2.1. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then there is a fibration

S^{3}{2^{n−1}} ⟶ ΩBG_{2}^{∧} ⟶ F_{n},

where F_{n} is a space of type S^{5}{2}.

The proof of theorem 4.2.1 will again require some group theoretic observations together with the an application of the techniques of [30]. The main idea of the proof is to obtain a non-explicit group homomorphism, p : K ⟶ G, for any finite 2-perfect group G, containing SD_{2^{n}} as a Sylow 2-subgroup, such that K is a finite 2-perfect subgroup of G, containing Q_{2^{n−1}} as a Sylow 2-subgroup.

Recall the following result from [40].

Theorem 4.2.2 (Martino). Let G be a finite group containing SD_{2^{n}} as a Sylow 2-subgroup. Then H^{∗}(G) is one of the following rings:

1. P [a_{3}, b_{4}, c_{5}]/(a^{2}b + c^{2})

2. P [a_{1}, b_{3}, c_{4}]/(a^{2}c + b^{2})

3. P [a_{1}, b_{3}, c_{4}, d_{5}]/(a^{3}, ab, ad, b^{2}c + d^{2})

4. P [x_{1}, y_{1}, u_{3}, p_{4}]/(x^{2} + xy, x^{3}, xu, u^{2} + p(x^{2} + y^{2}))

The following lemma is the analogue of lemma 4.1.1 above. Notice that the result is just partially the same.

Lemma 4.2.1. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Suppose that G contains a central element of order 2. Then G is not 2-perfect.

Proof. Suppose G contains a central element of order 2 and is 2-perfect. Let E < Z(G) be a subgroup of order 2. Choose a copy of SD_{2^{n}} in G, which contains E. Then E < Z(SD_{2^{n}}) = ⟨a^{2^{n−2}}⟩. Thus E = ⟨a^{2^{n−2}}⟩. Consider the central extension

0 ⟶ E ⟶ G ⟶ G/E ⟶ 1. (4.2.1)

In particular G/E contains D_{2^{n−1}} ≅ SD_{2^{n}}/⟨a^{2^{n−2}}⟩ as a Sylow 2-subgroup. Since G is 2-perfect, so is G/E. Thus by proposition 4.1.2, the 2-universal central extension L of G/E contains Q_{2^{n}} as a Sylow 2-subgroup. In particular, H^{2}(BL) = 0. Thus the projection BL ⟶ B(G/E) lifts to BG. On the level of Sylow 2-subgroups we get a commuting diagram

[figure: Q_{2^{n}}; SD_{2^{n}}; D_{2^{n−1}}]

where both maps into D_{2^{n−1}} are epimorphisms. This is, of course, impossible. Thus G is not 2-perfect.

Denote D = ⟨a^{2^{n−2}}, b⟩ < SD_{2^{n}}. Since a^{2^{n−2}} generates the center of SD_{2^{n}} and both a^{2^{n−2}} and b have order 2, it follows that D is elementary abelian.

Lemma 4.2.2. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then W_{G}D is isomorphic to a non-trivial subgroup of Σ_{3}.

Proof. This is essentially the same argument as in lemma 4.1.2.

Proposition 4.2.1. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then W_{G}D ≅ Σ_{3}

Proof. As in proposition 4.1.3, the proof is by contradiction. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup and suppose W_{G}(D) ≅ ℤ/2ℤ. Then, with the notation of lemma 4.2.2, it follows that the natural map ⟨a^{2^{n−3}}, b⟩/D ⟶ W_{G}(D) is an isomorphism. In particular, a representative for the generator α of W_{G}(D) is given by the element a^{2^{n−3}} ∈ SD_{2^{n}}. Thus one easily verifies that the operation of α on D is given by transposing the elements b and a^{2^{n−2}}b and fixing a^{2^{n−2}}.

Consider the subcategory Γ of A_{2}(G) whose objects are E = ⟨a^{2^{n−2}}⟩ and D, and whose morphisms are given by

Mor(D; D) ≅ W_{G}(D); Mor(E; E) = {id}; Mor(E; D) = {E ↪^{ι} D}.

The procedure described in the proof of proposition 4.1.3 can be followed verbatim in order to verify that Γ is a full subcategory of A_{2}(G). Furthermore, it is easy to see that that Γ is a proper Quillen subcategory of A_{2}(G).

Let α and ι be as above. Let

ᾱ : C_{G}(D) ⟶ C_{G}(D) ī: C_{G}(D) ⟶ C_{G}(E)

denote a choice of induced homomorphisms. By theorem 2.3.1, H^{∗}(BG) is given by the kernel of the map

ϕ : H^{∗}(BC_{G}(E)) × H^{∗}(BC_{G}(D)) ⟶ H^{∗}(BC_{G}(D)) × H^{∗}(BC_{G}(D)),

defined by ϕ(u, v) = (v−ī^{∗}u, v−ᾱ^{∗}v). Since C_{G}(E) contains a central element of order 2, it follows from lemma 4.2.1 that it is not 2-perfect. Let a_{1} ∈ H^{1}(C_{G}(E)) be a non-zero element. Since α fixes E, we have α ∘ i = i. Applying the (covariant) functor H^{∗}(BC_{G}(−)) we get ᾱ^{∗} ∘ ī^{∗} = ī^{∗}. Now ϕ(a_{1}, ī^{∗}(a_{1})) = (ī^{∗}a_{1} − ī^{∗}a_{1}, ī^{∗}a_{1} − ᾱī^{∗}a_{1}) = 0. Thus (a_{1}, ī^{∗}(a_{1})) represents a non-zero element of H^{1}(BG), contradicting 2-perfectness. This completes the proof.

Corollary 4.2.1. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Let Γ be the full subcategory of A_{2}(G) whose objects are D and all its non-trivial subgroups: E_{1} = ⟨a^{2n−2}⟩ ; E_{2} = ⟨b⟩ and E_{3} = ⟨ba^{2n−2}⟩. Then Γ is a proper Quillen subcategory of A_{2}(G) and the morphisms in Γ are given as follows: Mor(D; D) ≅ Σ_{3} generated by an element α of order 2 and an element γ of order 3; Mor(E_{i}; D) = All three possible injections of ℤ/2ℤ into D and Mor(E_{i}; E_{j}) = {β_{i,j}}, the unique non-zero map. Furthermore, commutativity relations in the category are given as follows: β_{k,j}β_{i,k} = β_{ij} and since Σ_{3} acts transitively on the non-unit elements of D, it follows that for every ψ ∈ Mor(E_{i}; D), there exists some ϵ ∈ Mor(D; D) such that ψ = ϵιβ_{i,1}, where ι : E_{1} ⟶ D is the natural inclusion.

Proof. This is straightforward from the definitions and proposition 4.2.1.

Proposition 4.2.2. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Let Γ be as in corollary 4.2.1. Let γ ∈ Mor(E_{i}; D) be any morphism. Then

H^{1}(BC_{G}(γ)) : H^{1}(BC_{G}(E_{i})) ⟶ H^{1}(BC_{G}(D))

is an injection.

Proof. From commutativity relations in the subcategory Γ of A_{2}(G), together with the fact that H^{∗}(BC_{G}(β_{j,k})) is an isomorphism for all (j, k), it follows that it suffices to check that the functor H^{1}(BC_{G}(−)) applied to the inclusion E_{1} ↪^{ι} D yields a monomorphism. For any morphism ψ in Γ, let δ(ψ) and ρ(ψ) denote the domain and range of ψ respectively. By theorem 2.3.1, H^{1}(BG) is given as the kernel of the map

ϕ : ∏_{i=1}^{3} H^{1}(BC_{G}E_{i}) × H^{1}(BC_{G}D) ⟶ ∏_{ψ∈MorΓ} H^{1}(BC_{G}(ρ(ψ))).

Denote a typical element in the domain of ϕ by

X = (X_{C_{G}E_{1}}, X_{C_{G}E_{2}}, X_{C_{G}E_{3}}, X_{C_{G}D}).

For any morphism ψ in Γ, let ϕ(X)_{ψ} denote the projection of ϕ(X) to its ψ-th coordinate. Then ϕ is given by the formula

ϕ(X)_{ψ} = X_{C_{G}(ρ(ψ))} − H^{1}(BC_{G}(ψ))X_{C_{G}(δ(ψ))}.

Let x ∈ Ker(H^{1}(BC_{G}(ι))) be any element. Consider the element

X = (x, H^{1}(BC_{G}(β_{1,2}))x, H^{1}(BC_{G}(β_{1,3}))x, 0) ∈ domain(ϕ).

Let ψ ∈ Mor(D; D), then clearly ϕ(X)_{ψ} = 0.

Let ψ ∈ Mor(E_{j}; D) for some j. Then

X_{C_{G}(ρ(ψ))} = 0 and X_{C_{G}(δ(ψ))} = X_{C_{G}(E_{j})} = H^{1}(BC_{G}(β_{1,j}))x.

Also for each i, j, the maps H^{∗}(BC_{G}(β_{i,j})) are isomorphisms with inverses given by H^{∗}(BC_{G}(β_{j,i})). Thus

ϕ(X)_{ψ} = H^{1}(BC_{G}(ψ))H^{1}(BC_{G}(β_{1,j}))x =
H^{1}(BC_{G}(ϵ))H^{1}(BC_{G}(ι))H^{1}(BC_{G}(β_{j,1}))H^{1}(BC_{G}(β_{1,j}))x =
H^{1}(BC_{G}(ϵ))H^{1}(BC_{G}(ι))x = 0,

where ϵ ∈ Mor(D; D) is some element such that ψ = ϵιβ_{j,1} (see corollary 4.2.1). Finally, let ψ = β_{i,j} ∈ Mor(E_{i}; E_{j}). Then

X_{C_{G}(ρ(ψ))} = X_{C_{G}(E_{j})} = H^{1}(BC_{G}(β_{1,j}))x

and

X_{C_{G}(δ(ψ))} = X_{C_{G}(E_{i})} = H^{1}(BC_{G}(β_{1,i}))x.

Also recall the relation β_{j,k}β_{i,j} = β_{i,k}. Thus

ϕ(X)_{ψ} = H^{1}(BC_{G}(β_{1,j}))x − H^{1}(BC_{G}(β_{i,j}))xH^{1}(BC_{G}(β_{1,i}))x = 0.

Consequently ϕ(X) = 0. Since G is assumed 2-perfect it follows that x = 0 so Ker(H^{1}(BC_{G}(ι))) = 0. This completes the proof.

Proposition 4.2.3. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then, with the notation of corollary 4.2.1,

H^{∗}(BC_{G}(E_{i})) ≅ P [a_{1}, b_{3}, c_{4}]/(a^{2}c + b^{2})

Proof. It suffices to prove the proposition for E = E_{1}. By lemma 4.2.1, C_{G}E is not 2-perfect. We show that if H^{∗}(BC_{G}(E)) has either structure 3 or 4 of theorem 4.2.2, then G cannot be 2-perfect. Indeed, since D is a central Sylow 2subgroup of C_{G}(D), the restriction Res(C_{G}(D), D) is an isomorphism by lemma 4.1.1. Let ι : E ⟶ D be as before. Then H^{1}(BC_{G}(ι)) is a monomorphism if and only if Res(C_{G}(E), D) is. Thus by proposition 4.2.2, it suffices to show that assuming either structure 3 or 4 for H^{∗}(BC_{G}(E)) implies that Res(C_{G}(E), D) has a non-trivial kernel.

We have Res(C_{G}E, D) = Res(SD_{2^{n}}, D)Res(C_{G}E, SD_{2^{n}}). With the notations of theorem 4.2.2, the restriction Res(SD_{2^{n}}, D) in dimension 1, is given by Res(x_{1}) = 0 and Res(y_{1}) = b̄, where b̄ ∈ H^{1}(BD) corresponds to the generator b of D.

By [40], if H^{∗}(BC_{G}E) has either structure 3 or 4 of theorem 4.2.2, then x_{1} ∈ Im(Res(C_{G}E, SD_{2^{n}})). Assuming either one of those structures, let z ∈ H^{1}(BC_{G}(E)) be an element such that Res(C_{G}(E), SD_{2^{n}})z = x_{1}. Then

Res(C_{G}(E), D)z = Res(SD_{2^{n}}, D)x_{1} = 0.

Hence by proposition 4.2.2, G is not perfect and so the structure of H^{∗}(BC_{G}E) is uniquely determined to be structure 2 of theorem 4.2.2. This completes the proof.

Proposition 4.2.4. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then there exists a 2-perfect subgroup K < G, containing the quaternion group Q_{2^{n−1}} as a Sylow 2-subgroup.

Proof. By proposition 4.2.3, H^{1}(BC_{G}E) ≅ ⟨a_{1}⟩ℤ/2ℤ and Sq^{1}a_{1} ≠ 0. Let ã_{1} : BC_{G}E ⟶ K(ℤ/2ℤ, 1) be a choice of a map corresponding to a_{1}. Then the homotopy fibre B̃ of a_{1} is equivalent to the classifying space of a normal subgroup K of C_{G}E. Furthermore, ã_{1} induces a mod-2 cohomology isomorphism through dimension 2. Thus K is 2-perfect. The Sylow 2-subgroup of K cannot be cyclic. Thus it must be isomorphic to either D_{2^{n−1}} or Q_{2^{n−1}}. By [25] Res(SD_{2^{n}}, E) is the zero map in dimension 1. Thus Res(C_{G}E, E)(a_{1}) = 0. It follows that the inclusion E ↪ C_{G}E factors through K. Since E is central in C_{G}E and K is a subgroup, we have that E is central in K. By lemma 4.1.1 the Sylow 2-subgroup of K cannot be dihedral. This completes the proof.

Recall that if K is a finite 2-perfect group containing Q_{2^{n−1}} as a Sylow 2subgroup then

H^{∗}(K) ≅ P [p_{4}] ⊗ E[u_{3}],

with β_{n−1}(u_{3}) = p_{4}.

Proposition 4.2.5. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Let K be a 2-perfect subgroup of G, containing Q_{2^{n−1}} as a Sylow 2-subgroup. Then Res(G, K) is an epimorphism.

Proof. Write

H^{∗}(BSD_{2^{n}}) ≅ P [x_{1}, y_{1}, u_{3}, p_{4}]/(x^{2} + xy, x^{3}, xu, u^{2} + p(x^{2} + y^{2})).

By [25], Res(SD_{2^{n}}, Q_{s^{n−1}})(p_{4}) = p̄_{4}, where p̄_{4} ∈ H^{4}(BQ_{2^{n−1}}) is the generator. Consider the diagram

[figure: Q_{2^{n−1}}; SD_{2^{n}}; K; G]

By [40], Res(G, SD_{2^{n}})(b_{4}) = p_{4} + y_{1}^{4}. Thus Res(G, Q_{2^{n−1}})(b_{4}) = p̄_{4}. Next, Res(K, Q_{2^{n}−1}) is an isomorphism in dimension 4. Thus Res(G, K)(b_{4}) = p_{4}. In H^{∗}(BG) we have β_{n−1}(a_{3}) = b_{4}, where in H^{∗}(BK), on the other hand, we have β_{n−1}(u_{3}) = p_{4}. Thus Res(G, K)(a_{3}) = u_{3}. This completes the proof.

Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Let K < G be a 2-perfect subgroup containing the quaternion group Q_{2^{n−1}} as a Sylow 2-subgroup, as in proposition 4.2.4. Let F_{n} be the homotopy fibre of the natural map BK_{2}^{∧} ⟶^{j} BG_{2}^{∧}. Then we get a commutative diagram of principal fibrations

[figure: A :; ΩBG_{2}^{∧}; =; F_{n}; BK_{2}^{∧}; j; δ; B :; ΩBG_{2}^{∧}; ∗; BG_{2}^{∧}] (4.2.2) which will eventually allow us to compute H^{∗}(F_{n}). First we need to compute H^{∗}(ΩBG_{2}^{∧}). This turns out to depend only on the cohomology ring of G itself and the action of the Steenrod algebra.

Proposition 4.2.6. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Then

H^{∗}(ΩBG_{2}^{∧}) ≅ Γ[z_{8}] ⊗ P [x_{2}]/(x^{4}) ⊗ E[y_{3}]
H_{∗}(ΩBG_{2}^{∧}) ≅ P [z_{4}] ⊗ E[u_{2}, v_{3}]

as algebras, where degrees of generators are given by their subscripts. Moreover, the Hopf algebra structure of H_{∗}(ΩBG_{2}^{∧}) is determined by the requirement that u and v are primitive and Δ(z) = 1 ⊗ z + z ⊗ 1 + u ⊗ u.

Proof. Recall

H^{∗}(BG) ≅ P [a_{3}, b_{4}, c_{5}]/(a^{2}b + c^{2}).

Denote H^{∗}(BG) by Λ. Then Λ is a free module over P [a_{3}, b_{4}]. Indeed the elements a_{3} and b_{4} generate a Borel ideal in Λ, and Λ/(a_{3}, b_{4}) ≅ E[c_{5}]. Thus there is a spectral sequence

(4.2.3) E_{2} = Tor_{E[c_{5}]}(𝔽_{2}; Tor_{P [a_{3},b_{4}]}(𝔽_{2}; 𝔽_{2})) ⟹ Tor_{Λ}(𝔽_{2}; 𝔽_{2})

But

Tor_{E[c_{5}]}(𝔽_{2}; Tor_{P [a_{3},b_{4}]}(𝔽_{2}; 𝔽_{2})) ≅
Tor_{E[c_{5}]}(𝔽_{2}; 𝔽_{2}) ⊗ Tor_{P [a_{3},b_{4}]}(𝔽_{2}; 𝔽_{2}) ≅
Γ[(c_{5}, −1)] ⊗ E[(a_{3}, −1), (b_{4}, −1)].

One verifies by direct calculation that all differentials in the spectral sequence vanish. Thus

E_{2} ≅ E_{∞} ≅ E_{0}(Tor_{Λ}(𝔽_{2}; 𝔽_{2})) ≅ Γ[(c_{5}, −1)] ⊗ E[(a_{3}, −1), (b_{4}, −1)].

Notice that this isomorphism respects the filtration induced on Tor_{Λ}(𝔽_{2}; 𝔽_{2}) by filtering Λ by the powers of its augmentation ideal.

Next we have the Eilenberg-Moore spectral sequence with

E_{2} ≅ Tor_{Λ}(𝔽_{2}; 𝔽_{2}) ⟹ H^{∗}(ΩBG_{2}^{∧}).

This last spectral sequence can be viewed as the one arising from filtering Λ by the powers of its augmentation ideal. Thus by degree reasons there cannot be any non-trivial differentials, and so, the spectral sequence degenerates and there is an isomorphism

E_{0}(H^{∗}(ΩBG_{2}^{∧})) ≅ Γ[c_{4}] ⊗ E[x_{2}, y_{3}].

The dual computation yields

E^{0}(H_{∗}(ΩBG_{2}^{∧})) ≅ P [z_{4}] ⊗ E[u_{2}, v_{3}].

Notice in particular that H^{n}(ΩBG_{2}^{∧}) ≅ ℤ/2ℤ for every n ≥ 2.

We next claim that

H_{∗}(ΩBG_{2}^{∧}) ≅ E^{0}(H_{∗}(ΩBG_{2}^{∧})) ≅ P [z_{4}] ⊗ E[u_{2}, v_{3}]

as algebras and that the Hopf algebra structure on H_{∗}(ΩBG_{2}^{∧}) is determined by the requirement that Δ(z_{4}) = z_{4} ⊗ 1 + 1 ⊗ z_{4} + u_{2} ⊗ u_{2}.

For simplicity of notation denote the generators of E^{0}(H_{∗}(ΩBG_{2}^{∧})) by u, v and z. Consider the cohomology Serre spectral sequence for the path-loop fibration over BG_{2}^{∧}. We must have d_{3}(x_{2}) = a_{3}. Hence

d_{5}(x_{2}^{2}) = d_{5}(Sq^{2}x_{2}) = Sq^{2}d_{3}(x_{2}) = Sq^{2}(a_{3}) = c_{5} = d_{5}(c_{4}).

Therefore, in H^{∗}(ΩBG_{2}^{∧}), we have x_{2}^{2} = c_{4}. Similarly x_{2}^{3} = x_{2}c_{4} ≠ 0. However, d_{9}(x_{2}^{4}) = Sq^{4}Sq^{2}d_{3}(x_{2}) = Sq^{4}Sq^{2}(a_{3}) = Sq^{4}(c_{5}) = ϵa_{3}^{3} + δb_{4}c_{5} = 0 in E_{9}. Thus x_{2}^{4} = 0 and the subalgebra of H^{∗}(ΩBG_{2}^{∧}) generated by x_{2} is isomorphic to P [x_{2}]/(x^{4}).

Dually in H_{∗}(ΩBG_{2}^{∧}) we have Δ(z) = z ⊗ 1 + 1 ⊗ z + u ⊗ u. We proceed by computing the product structure in H_{∗}(ΩBG_{2}^{∧}). First observe that u and v must be primitive for dimension reasons. Thus u^{2} and v^{2} are also primitive. Since z and uz are not primitive and are the unique generators in dimensions 4 and 6 respectively, it follows that u^{2} = v^{2} = 0.

Next uv = ϵ·vu, ϵ ∈ 𝔽_{2}. If ϵ = 0 then uv = 0. But Δ̃(uv) = u⊗v +v ⊗u ≠ 0. This yields a contradiction and thus implies uv = vu. We thus conclude that the commutative algebra E[u, v] is contained in H_{∗}(ΩBG_{2}^{∧}) as a subalgebra.

Consider the products uz and vz. We have uz = ϵ · zu and vz = δ · zv, ϵ, δ ∈ 𝔽_{2}. If either ϵ or δ is equal to zero then the same argument as above yields a contradiction. Thus we conclude that z commutes with every element of E[u, v].

Finally, since z has infinite height in E^{0}(H_{∗}(ΩBG_{2}^{∧})), the same is true in H_{∗}(ΩBG_{2}^{∧}). Since the associated graded is generated by u, v and z, this completes the computation of H_{∗}(ΩBG_{2}^{∧}) as a Hopf algebra. The result for H^{∗}(ΩBG_{2}^{∧}) follows by duality.

Remark 4.2.1. Consider the cohomology Serre spectral sequence for the pathloop fibration over BG_{2}^{∧}. We have:

E_{2} ≅ P [a_{3}, b_{4}, c_{5}]/(a^{2}b + c^{2}) ⊗ (P [x_{2}]/(x^{4}) ⊗ E[y_{3}] ⊗ Γ[z_{8}])

For dimension reasons we must have d_{3}(x) = a, d_{4}(y) = b and d_{5}(x^{2}) = c. One now easily observes that this also forces d_{5}(γ_{i}(z)) = x^{2}γ_{i−1}(z) · c.

Proposition 4.2.7. H^{∗}(F_{n}) ≅ Γ[υ_{4}] ⊗ E[μ_{5}] with Sq^{1}(γ_{i}(υ)) = μγ_{i−1}(υ)

Proof. Consider diagram 4.2.2 above. The cohomology Serre spectral sequence for A has the form

E_{2}(A) ≅ H^{∗}(BK_{2}^{∧}) ⊗ H^{∗}(ΩBG_{2}^{∧}) ≅
(P [p_{4}] ⊗ E[u_{3}]) ⊗ (P [x_{2}]/(x^{4}) ⊗ E[y_{3}] ⊗ Γ[z_{8}]).

By proposition 4.2.5, j^{∗} is an epimorphism. Thus

j^{∗}(a_{3}) = u_{3}, j^{∗}(b_{4}) = p_{4}, and j^{∗}(c_{5}) = 0.

By remark 4.2.1 and naturality, we have in E_{∗}(A)

d_{3}(x_{2}) = u_{3}, d_{3}(x_{2}u_{3}) = 0, d_{4}(y_{3}) = p_{4}, and d_{5}(x_{2}^{2}) = 0.

This determines the spectral sequence. We have

E_{4}(A) ≅ P [p_{4}] ⊗ E[x_{2}u_{3}] ⊗ (P [x_{2}^{2}]/(x^{4}) ⊗ E[y_{3}] ⊗ Γ[z_{8}])

and

E_{5}(A) ≅ E[x_{2}u_{3}] ⊗ (P [x_{2}^{2}]/(x^{4}) ⊗ Γ[z_{8}]) ≅ E_{∞}(A).

Notice that P [x_{2}^{2}]/(x^{4}) ⊗ Γ[z_{8}] ≅ Γ[υ_{4}] as algebras over 𝔽_{2}. Thus

E_{0}(H^{∗}(F_{n})) ≅ Γ[υ_{4}] ⊗ E[μ_{5}].

Observe that the restriction to H^{∗}(ΩBG_{2}^{∧}) maps the submodule of H^{∗}(F_{n}) corresponding to the subalgebra Γ[υ_{4}] of E_{0}(H^{∗}(F_{n})) isomorphically into the subalgebra P [x_{2}^{2}]/(x^{4}) ⊗ Γ[z_{8}] of H^{∗}(ΩBG_{2}^{∧}). This subalgebra is itself isomorphic to a divided polynomial algebra on one generator in dimension 4. Thus H^{∗}(F_{n}) contains Γ[υ_{4}] as a subalgebra. By dimension reasons, we have μ_{5}^{2} = 0. Thus the algebra structure of H^{∗}(F_{n}) is determined. Notice that in homology we have the subring P [z_{4}] of H_{∗}(ΩBG_{2}^{∧}) injecting into H_{∗}(F_{n}). In fact, since the fibration A above is principal that makes H_{∗}(F_{n}) into a free P [z_{4}] module on two generators 1 and u_{5}. Thus we may write the generators of H_{∗}(F_{n}) in the form z_{4}^{i} · 1 or z_{4}^{i} · u_{5}.

Next consider the cohomology Serre spectral sequence for the fibration

F_{n} ⟶ BK_{2}^{∧} ⟶ BG_{2}^{∧}.

Since the cohomology algebras of all three spaces are known, one immediately concludes that in the spectral sequence, d_{5}(υ_{4}) = c_{5}. Thus d_{6}Sq^{1}(υ_{4}) = Sq^{1}d_{5}(υ_{4}) = Sq^{1}(c_{5}) = a_{3}^{2}. Consequently we get Sq^{1}(υ_{4}) ≠ 0, so Sq^{1}(υ_{4}) = μ_{5}. Dually, in homology, Sq_{∗}^{1}(u_{5}) = z_{4} ·1. Thus Sq_{∗}^{1}(z_{4}^{i} ·u_{5}) = z_{4}^{i} ·Sq_{∗}^{1}(u_{5}) = z_{4}^{i} ·z_{4} ·1 = z_{4}^{i+1} ·1. Duallizing again we get Sq^{1}(γ_{i+1}(υ_{4})) = μ_{5}γ_{i}(υ_{4}). This completes the proof.

We are now ready to prove theorem 4.2.1. By proposition 4.2.7, H^{∗}(F_{n}) ≅ H^{∗}(S^{5}{2}) as algebras over the Steenrod algebra. We have a fibration

ΩBK_{2}^{∧} ⟶ ΩBG_{2}^{∧} ⟶ F_{n}

where K is a subgroup of G containing Q_{2^{n−1}} as a Sylow 2-subgroup. By proposition 4.1.1, ΩBK_{2}^{∧} ≃ S^{3}{2^{n−1}}. This completes the proof.

Remark 4.2.2. One might wonder whether the spaces F_{n} are homotopy equivalent to S^{5}{2} or whether they are spherically resolvable in general.

Remark 4.2.3. Let G be a finite 2-perfect group containing SD_{2^{n}} as a Sylow 2-subgroup. Using different techniques than the ones described here, one can show that ΩBG_{2}^{∧} is S-resolvable [19].

84 4. SPORADIC EXAMPLES

CHAPTER 5 Groups of Lie Type and S-Resolutions

In this chapter we show that the classifying spaces of finite groups of Lie type completed at most primes have S-resolvable loop spaces.

5.1. Preliminary Theorems

Let G be a complex connected reductive Lie group. Let G_{ℤ} be an associated integral Chevalley group scheme. Let K be an algebraically closed field of characteristic l > 0 and let G_{K} be the algebraic group associated with G_{Z} over K. For any group endomorphism ϕ : G ⟶ G, let E(ϕ) < G_{K} be the subgroup consisting of all K-rational points in G_{K} invariant under the action of ϕ.

If X, Y_{1} and Y_{2} are any topological spaces and f_{i} : X ⟶ Y_{i}, i = 1, 2 are maps, then we denote by f_{1}⊤f_{2} : X ⟶ Y_{1} × Y_{2} the canonical map into the product determined by f_{1} and f_{2}.

Recall from [27]

Theorem 5.1.1 (Friedlander). Let G be a complex connected reductive Lie group. Let ϕ : G_{K} ⟶ G_{K} be a surjective endomorphism such that the subgroup E(ϕ) < G_{K} is finite. Then for every prime p ≠ l there is a homotopy pull back square

[figure: BE(ϕ)_{p}^{∧}; D; BG_{p}^{∧}; Δ; BG_{p}^{∧}; (BG_{p}^{∧})^{×2}; 1⊤ϕ] (5.1.1)

where Δ is the diagonal. Furthermore, the square is natural with respect to maps induced by morphisms of integral Chevalley group schemes.

86 5. GROUPS OF LIE TYPE AND S-RESOLUTIONS

Example 5.1.1. Let K = 𝔽̄_{l}, the algebraic closure of the prime field of l elements. Let q = l^{k}. Let ϕ^{q} : K ⟶ K be the Frobenius automorphism taking an element to its q-th power. Let G(ℂ) be a complex reductive Lie group. Let G_{K} be an associated Chevalley group scheme over K. Let G(K) be the corresponding group of K-rational points in G_{K}. Then ϕ^{q} induces an automorphism ϕ^{q} : G(K) ⟶ G(K) whose invariant subgroup is the finite group G(𝔽_{q}) of 𝔽_{q} rational points in G_{K}. Thus in this case theorem 5.1.1 claims that for p ≠ l, BG(𝔽_{q})_{p}^{∧} is obtained by means of the pull-back square (5.1.1) from the p-completed classifying space of the corresponding complex Lie group. Furthermore, since ϕ^{q} is induced by a field automorphism, it commutes with any map which arises from a morphism of integral Chevalley group schemes.

Complex Lie groups are in particular finite H-spaces. The Borel structure theorem [9] on the cohomology of finite H-spaces turns out to be useful for our purpose.

Theorem 5.1.2 (Borel). Let X be a finite H-space such that

H^{∗}(X; ℚ) ≅ E[x_{1}, x_{2}, ⋯, x_{r}].

Then

H^{∗}(X; ℤ)/T orsion ⊗ 𝔽_{p} ≅ E[x_{1}, x_{2}, ⋯, x_{r}].

In particular if p is a prime such that H^{∗}(X; ℤ) does not have p-torsion then H^{∗}(X; 𝔽_{p}) ≅ E[x_{1}, x_{2}, ⋯, x_{r}].

Finally we recall from [21]

Theorem 5.1.3 (Cooke, Harper, Zabrodsky). Let X be a 1-connected plocal H-space, such that H^{∗}(X; ℤ) is torsion free and

H^{∗}(X; 𝔽_{p}) ≅ E[x_{1}, x_{2}, ⋯, x_{r}]; |x_{i}| = 2m_{i} − 1

with r < p − 1. Then there exists a map f : X ⟶ (S^{2m_{r}−1})_{(p)} whose homotopy fibre Y is a simply connected p-local H-space with torsion free integral cohomology and such that H^{∗}(Y; 𝔽_{p}) ≅ E[x_{1}, x_{2}, ⋯, x_{r−1}].

5.2. A Spherical Fibre Square

We imitate the construction of Friedlander in Theorem 5.1.1 in terms of odd dimensional spheres rather than complex Lie groups. Let p be a prime number and q a positive integer. Let r = ν_{p}(q − 1). We claim that if p is odd then, for every n ≥ 1, the homotopy fibre S^{2n+1}{p^{r}} of the degree p^{r} self map of S^{2n+1} can be obtained by means of a pull back square similar to that of theorem 5.1.1. Recall that a simply connected p-local space X is said to be of type S^{2n+1}{p^{r}} if H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(S^{2n+1}{p^{r}}; 𝔽_{p}) as an algebra over the Steenrod algebra with the same secondary Bockstein operators. For p = 2, we show that the corresponding pull-back square yields a space S(2n+1, 2^{r}) of type S^{2n+1}{2^{r}}, which is, in particular, S-resolvable. Notice that we do not exclude r = 0, in which case both S^{2n+1}{1} and S(2n + 1, 1) are contractible. Throughout this section we assume that all spaces and maps are p-completed for some fixed prime p.

Proposition 5.2.1. Let p be an odd prime and q > 1 an integer. Let r = ν_{p}(q − 1). Then for every n ≥ 1 there is a homotopy pull back square

[figure: S^{2n+1}{p^{r}}; S^{2n+1}; Δ; S^{2n+1}; (S^{2n+1})^{×2}; 1⊤[q]]

where Δ is the diagonal map and [q] is a degree q self map of S^{2n+1}.

Proof. Since p is odd, S^{2n+1} is an H-space. Let μ̄ : S^{2n+1}×S^{2n+1} ⟶ S^{2n+1} denote the composite

S^{2n+1} × S^{2n+1} ⟶^{[−1]×[1]} S^{2n+1} × S^{2n+1} ⟶^{μ} S^{2n+1},

where μ is the H-space structure map. Let F be the homotopy fibre of μ̄. Then there is a principal fibration

ΩS^{2n+1} ⟶^{j} F ⟶^{π} S^{2n+1} × S^{2n+1}.

The associated ℤ_{(p)}-cohomology Serre spectral sequence takes the form

E_{2}^{∗,∗} ≅ H^{∗}(S^{2n+1} × S^{2n+1}; ℤ_{(p)}) ⊗ H^{∗}(ΩS^{2n+1}; ℤ_{(p)}) ≅ E[x, y] ⊗ Γ[γ(z)],

where |x| = |y| = 2n + 1, |z| = 2n and d(γ_{1}(z)) = y − x. Since the fibration is multiplicative, this determines the rest of the differentials. We have d(γ_{k}(z)) = (y − x)γ_{k−1}(z) and d(xγ_{k}(z)) = xyγ_{k−1}(z). Thus

E_{2n+2}^{p,q} ≅ E_{∞}^{p,q} ≅ \{ \begin{matrix}ℤ & p = 0 or 2n + 1, q = 0 \\ 0 & otherwise. \end{matrix}

This determines the homotopy type of F to be that of S^{2n+1} and the projection π is easily seen to be the diagonal map, up to homotopy.

Next we observe that the composite

S^{2n+1} ⟶^{1⊤[q]} S^{2n+1} × S^{2n+1} ⟶^{μ̄} S^{2n+1}

is a degree (q − 1) map. Thus we get a homotopy commutative square

[figure: S^{2n+1}; [q−1]; (S^{2n+1})^{×2}; μ̄; 1⊤[q]; S^{2n+1}; S^{2n+1}; =] (5.2.1)

Since all spaces are p-local, [q − 1] is homotopic to the composition of [p^{r}] with a self equivalence of S^{2n+1}. Thus the fibre of [q − 1] is equivalent to S^{2n+1}{p^{r}}. Taking fibres on (5.2.1) now gives the required fibre square.

We conclude this section by discussing the statement corresponding to proposition 5.2.1 for p = 2. First need the following well known lemma.

Lemma 5.2.1. Let X be a simply connected space. Let f : X ⟶ X be any self-map. Then the homotopy fibre of the map 1⊤f : X ⟶ X × X is homotopy equivalent to ΩX.

Proof. There is a commutative diagram

[figure: X; =; X × X; p_{1}; 1⊤f; X; X; =]

where p_{1} denotes the projection to the first factor. By [20], one can extend this square into a 3×3 diagram where all rows and columns are fibration sequences. The result is immediate.

In the following proposition we fix p = 2. Let q be a positive integer and r = ν_{2}(q − 1). Define S(2n + 1, 2^{r}) to be the pull-back in the diagram

S^{2n+1} ⟶^{1⊤[q]} S^{2n+1} × S^{2n+1} ⟵^{Δ} S^{2n+1}.

Proposition 5.2.2. For every n ≥ 1 the space S(2n+1, 2^{r}) is a space of type S^{2n+1}{2^{r}}.

Proof. By lemma 5.2.1, we have a homotopy commutative diagram

[figure: A :; ΩS^{2n+1}; =; S(2n + 1, 2^{r}); S^{2n+1}; 1⊤[q]; B :; ΩS^{2n+1}; S^{2n+1}; (S^{2n+1})^{×2}; Δ]

where the rows are fibrations. Write H_{∗}(ΩS^{2n+1}; ℤ_{2}^{∧}) ≅ P [v_{2n}]. Let

x_{2n+1}, y_{2n+1}, z_{4n+2} ∈ H_{∗}((S^{2n+1})^{×2}; ℤ_{2}^{∧}) and w_{2n+1} ∈ H_{∗}(S^{2n+1}; ℤ_{2}^{∧})

denote the generators.

In the ℤ_{2}^{∧}-homology Serre spectral sequence for B we have d(x) = −d(y) = v up to a sign. Thus in the spectral sequence for A we have, by naturality, d(w) = (q − 1)v, up to a sign. Since coefficients are taken to be ℤ_{2}^{∧}, we have (q − 1) = 2^{r} · ζ, where ζ ∈ ℤ_{2}^{∧} is a unit. Thus we may assume d(w) = 2^{r} · v. We now proceed in computing the ℤ_{2}^{∧}-homology Serre spectral sequence for A. We have

E^{2} ≅ H_{∗}(S^{2n+1}; ℤ_{2}^{∧}) ⊗ H_{∗}(ΩS^{2n+1}; ℤ_{2}^{∧})

as a differential graded coalgebra. We abbreviate w ⊗ 1 by w, 1 ⊗ v^{k} by v^{k} and w ⊗ v^{k} by wv^{k}. We must compute d(wv^{k}) for all k. Since the spectral sequence is a spectral sequence of coalgebras, we have

Δd(wv) = (1 ⊗ d + d ⊗ 1)Δ(wv) = 1 ⊗ d(wv) + d(wv) ⊗ 1 + 2^{r+1}(v ⊗ v).

On the other hand,

Δ(v^{2}) = 1 ⊗ v^{2} + v^{2} ⊗ 1 + 2(v ⊗ v).

Thus d(wv) = 2^{r}v^{2}. By induction we conclude that d(wv^{k}) = 2^{r}v^{k+1}. It follows that the map

H_{∗}(ΩS^{2n+1}; ℤ_{2}^{∧}) ⟶ H_{∗}(S(2n + 1, 2^{r}); ℤ_{2}^{∧})

is just the reduction mod 2^{r}. The result follows.

Remark 5.2.1. Notice in particular that the spaces S(2n+1, 2^{r}) of proposition 5.2.2 are S-resolvable, a property they share with S^{2n+1}{2^{r}}. Whether or not S^{2n+1}{2^{r}} ≃ S(2n + 1, 2^{r}) remains an open question.

5.3. Proof of Theorem 1.1.3

We say that G is p-simply-connected if G_{p}^{∧} is simply connected.

Proposition 5.3.1. Let G be a complex connected reductive Lie group, which is p-simply-connected. Suppose that G_{p}^{∧} is finitely S-resolvable. Then so is ΩBG(𝔽_{q})_{p}^{∧} for any q with (q, p) = 1.

Proof. This is immediate by theorem 5.1.1 and lemma 3.2.1.

Definition 5.3.1. Let G be a complex connected reductive Lie group and p be a prime number. Let G_{Z} be an associated integral Chevalley group scheme, such that G = G(ℂ), the group of ℂ-rational points in the complex Chevalley group scheme G_{C} corresponding to G_{Z}. G is said to fibre algebraically over an m-dimensional sphere at the prime p with fibre H, if it is p-simply-connected and there exists a p-simply-connected Lie subgroup H of G such that the inclusion H ↪ G is induced by a map of integral Chevalley group schemes and the orbit space (G/H)_{p}^{∧} is homotopy equivalent to (S^{m})_{p}^{∧}. The group G is said to be algebraically resolvable over k spheres of dimensions m_{1}, m_{2}, ⋯ m_{k}, at the prime p, if it is p-simply-connected and there exists a sequence of p-simply-connected subgroups

G = G_{k} > G_{k−1} > G_{k−2} > ⋯ > G_{1} > G_{0} = {1},

such that G_{1} is equivalent to S^{m_{1}} at the prime p and for i = 2 ⋯ k, each G_{i} fibres algebraically over an m_{i}-dimensional sphere at the prime p with fibre G_{i−1}.

Let G be a complex connected reductive Lie group which fibres algebraically over an m-dimensional sphere at a prime p, with fibre H. Then we have a fibration

(S^{m})_{p}^{∧} ⟶ BH_{p}^{∧} ⟶ BG_{p}^{∧}.

Let q = l^{k} for some prime number l ≠ p. Since the map BH_{p}^{∧} ⟶ BG_{p}^{∧} is induced by a morphism of integral Chevalley group schemes, it commutes with the Frobenius homomorphism ϕ^{q}. Thus ϕ^{q} induces a self map on the fibre (S^{m})_{p}^{∧}, which is determined, up to homotopy, by its degree. We are now ready to prove theorem 1.1.3, which we restate as

Theorem 5.3.1. Let G be a complex connected reductive Lie group which is algebraically resolvable at the prime p, over k odd spheres of dimensions 2m_{i} + 1, i = 1 ⋯ k. Let G_{1}, ⋯, G_{k} be a sequence of subgroups of G associated with the resolution. Let q = l^{n} for some prime l ≠ p and let G_{i}(𝔽_{q}) be the associated finite Chevalley groups. Let t_{i} = ν_{p}(s_{i} − 1), where s_{i} is the degree of the map induced on S^{2m_{i}+1} by ϕ^{q}. Then the groups G_{i}(𝔽_{q}) are p-perfect and for every i = 1 ⋯ k there are fibrations

ΩBG_{i−1}(𝔽_{q})_{p}^{∧} ⟶ ΩBG_{i}(𝔽_{q})_{p}^{∧} ⟶ S_{i},

where S_{i} ≃ S^{2m_{i}+1}{p^{t_{i}}} if p is odd and S_{i} ≃ S(2m_{i} + 1, 2^{t_{i}}) if p = 2. Thus for every 1 ≤ i ≤ k, the space ΩBG_{i}(𝔽_{q})_{p}^{∧} is S-resolvable.

Proof. By hypothesis, the inclusions G_{i} ↪ G_{i+1} satisfy the conditions of theorem 5.1.1. Thus we get a commutative cube

[figure: BG_{i−1}(𝔽_{q})_{p}^{∧}; BG_{i−1}; 1⊤ϕ^{q}; BG_{i−1}; BG_{i−1}^{×2}; BG_{i}(𝔽_{q})_{p}^{∧}; BG_{i}; 1⊤ϕ^{q}; BG_{i}; BG_{i}^{×2}; Δ] (5.3.1)

where all spaces are taken to be p-completed and the top and bottom faces are homotopy pull back squares. Let S_{i} be the homotopy theoretic fibre of the map BG_{i−1}(𝔽_{q})_{p}^{∧} ⟶ BG_{i}(𝔽_{q})_{p}^{∧}. Then by lemma 2.4.1, the square of fibres of diagram (5.3.1) takes the form

[figure: S_{i}; S^{2m_{i}+1}; Ψ; S^{2m_{i}+1}; (S^{2m_{i}+1})^{×2}; Φ]

and is a homotopy pull back square. the map Ψ is induced by Δ and Φ is induced by 1⊤ϕ^{q}. The maps Φ and Ψ are determined by their degrees, thus clearly Ψ is homotopic to the diagonal map and Φ to 1⊤[s_{i}]. Since all spaces are p-local, it follows from propositions 5.2.1 and 5.2.2 that S_{i} ≃ S^{2m_{i}+1}{p^{t_{i}}} if p is odd S_{i} ≃ S(2m_{i} + 1, 2^{t_{i}}) if p = 2. This completes the proof.

The powers t_{i} in theorem 1.1.3 can sometimes be computed explicitly, as we demonstrate in the proof of theorems 1.1.4 and 1.1.5 below. We end this section with an observation which we shall use in the computation.

Let G be a complex connected Lie group. Let Ḡ < G be a maximal compact subgroup. Let T < Ḡ be a maximal torus. Suppose that p is a prime number such that H^{∗}(BḠ; ℤ) does not have p-torsion. Then, by [9], the restriction map H^{∗}(BḠ; ℤ_{(p)}) ⟶ H^{∗}(BT; ℤ_{(p)}) is a monomorphism. The inclusion ī : T ↪ Ḡ corresponds to a map of integral Chevalley group schemes i : GL_{1,ℤ}^{×n} ⟶ G_{ℤ}, where n is the rank of T. Thus for every prime power q, the map Bi_{p}^{∧} commutes with ϕ^{q}. Notice that (Bϕ^{q})_{p}^{∧} : (BGL_{1}(ℂ)^{×n})_{p}^{∧} ⟶ (BGL_{1}(ℂ)^{×n})_{p}^{∧} induces multiplication by q on the generators of H^{∗}(BGL_{1}(ℂ)^{×n})_{p}^{∧}; ℤ_{(p)}) [35]. Using these facts together with Borel’s theorem, one concludes the following lemma. The reader is referred to [35] for details.

Lemma 5.3.1. Let G be a complex connected reductive Lie group. Let p be a prime number such that H^{∗}(BG; ℤ) does not have p-torsion. Let q = l^{k} be some prime power with l ≠ p. Then for every x ∈ H^{2m}(BG; ℤ_{(p)}), (ϕ^{q})^{∗}(x) = q^{m}x.

5.4. The Groups SL_{n}(𝔽_{q}) and Sp_{2n}(𝔽_{q})

Let R be a commutative ring with a unit. Let SL_{n}(R) denote the special linear group of rank n over R. The integral algebraic group SL_{n,ℤ} is represented by its coordinate ring

SL_{n} = ℤ[x_{1,1} ⋯ x_{n,n}]/(det(x_{i,j}) − 1 = 0).

The natural inclusions SL_{n−1}(R) ⟶ SL_{n}(R) correspond to the ring homomorphisms p_{n} : SL_{n} ⟶ SL_{n−1} given by

p_{n}(x_{i,j}) = \{ \begin{matrix}x_{i,j} & i, j ≠ n \\ 0 & i = n and j ≠ n or i ≠ n and j = n \\ 1 & i = j = n. \end{matrix}

Thus, in particular, for R = ℂ, the complex numbers, we have that the natural map p_{n}^{∗} : BSL_{n−1}(ℂ) ⟶ BSL_{n}(ℂ) is induced by a morphism of integral Chevalley group schemes. Notice that the homotopy fibre of p_{n}^{∗} is equivalent to the sphere S^{2n−1}.

The group SL_{n}(ℂ) contains the special unitary group SU(n) as a maximal compact subgroup and the natural inclusion SU(n) ↪ SL_{n}(ℂ) is a homotopy equivalence. Furthermore, an inclusion of a maximal torus (S^{1})^{×(n−1)} ↪ SU(n) corresponds to the natural map GL_{1}(ℂ)^{×(n−1)} ⟶ SL_{n}(ℂ) given by sending an (n − 1)-tuple (z_{1}, z_{2}, ⋯, z_{n−1}) to the diagonal matrix

diag(z_{1}, z_{2}, ⋯, z_{n−1}, ∏_{i=1}^{n} z_{i}^{−1}).

We are now ready to prove theorem 1.1.4. Since H^{∗}(BSU(n); ℤ) is torsionfree, the conditions of lemma 5.3.1 are satisfied. Thus write

H^{∗}(BSL_{n}(ℂ); ℤ) ≅ P [w_{4}, w_{6}, ⋯ w_{2n}].

Then (ϕ^{q})^{∗}(w_{2i}) = q^{i}w_{2i}. Finally, we have a commutative diagram of fibrations

[figure: (S^{2n−1})_{p}^{∧}; s_{n}; BSL_{n−1}(ℂ)_{p}^{∧}; ϕ^{q}; BSL_{n}(ℂ)_{p}^{∧}; ϕ^{q}; (S^{2n−1})_{p}^{∧}; BSL_{n−1}(ℂ)_{p}^{∧}; BSL_{n}(ℂ)_{p}^{∧}]

In the associated integral cohomology Serre spectral sequence, the fundamental class x_{2n+1} ∈ H^{∗}(S^{2n−1}; ℤ) transgresses to w_{2n}. Thus the map s_{n} has degree q^{n}. The result follows from theorem 1.1.3.

Our next example is the symplectic groups Sp_{2n}(𝔽_{q}). The complex symplectic group Sp_{2n}(ℂ) contains the symplectic group of rank n over the quaternion algebra, which we denote by Sp_{n}, as a maximal compact subgroup. For every n ≥ 1 there are fibrations

Sp_{n−1} ⟶ Sp_{n} ⟶ S^{4n−1}.

In particular, the integral cohomology of BSp_{n} is torsion free. Thus, the method employed in the proof of theorem 1.1.4 works in this case as well. Theorem 1.1.5 follows.

5.5. Proof of Theorem 1.1.6 and Examples

Let G be a Lie group satisfying the hypotheses of the theorem. Then the group G and the primes p and l satisfy the conditions of Friedlander’s theorem 5.1.1. Thus there is a fibration

ΩG_{p}^{∧} ⟶ ΩBG(𝔽_{q})_{p}^{∧} ⟶ G_{p}^{∧}.

Thus by corollary 3.2.1, it suffices to show that G_{p}^{∧} is S-resolvable. But this follows at once from theorems 5.1.2 and 5.1.3.

Theorem 1.1.6 is not interesting for a fixed q and a sufficiently large prime p, since in that case the group G(𝔽_{q}) might not have a non-trivial p-subgroup and so ΩBG(𝔽_{q})_{p}^{∧} is contractible. However, since q is allowed to be an arbitrary prime power, the theorem does provide non-trivial examples of S-resolvable spaces of the form ΩBG_{p}^{∧}, for G finite and p-perfect group of Lie type.

Example 5.5.1. Consider the exceptional complex Lie group G_{2}. The cohomology ring H^{∗}(G_{2}; ℤ) is known to have only 2-primary torsion. Also the rank of H^{∗}(G_{2}; ℚ) is 2. Thus for p ≥ 5 and q = l^{k}, l ≠ p, the space ΩBG_{2}(𝔽_{q})_{p}^{∧} is S-resolvable of length at most 4. In particular the order of G_{2}(𝔽_{q}) is q^{6}(q^{6} − 1)(q^{2} − 1). Thus for example |G_{2}(𝔽_{9})| = 2^{8} · 3^{12} · 5^{2} · 7 · 13 · 73 and so ΩBG_{2}(𝔽_{9})_{p}^{∧} is S-resolvable at 5, 7, 13, and 73.

Example 5.5.2. The cohomology ring of the exceptional complex Lie group F_{4} is known to have only 2 and 3 torsion. The rank of H^{∗}(F_{4}; ℚ) is 4. Thus for p ≥ 7 and q = l^{k}, l ≠ p, the space ΩBF_{4}(𝔽_{q})_{p}^{∧} is S-resolvable of length at most 8. In particular the order of F_{4}(𝔽_{q}) is q^{24}(q^{12} − 1)(q^{8} − 1)(q^{6} − 1)(q^{2} − 1). Thus for example |F_{4}(𝔽_{3})| = 2^{15} · 3^{24} · 5^{2} · 7^{2} · 13^{2} · 41 · 73 ans so ΩBF_{4}(𝔽_{3})_{p}^{∧} is S-resolvable at 7, 13, 41, and 73.

CHAPTER 6 Clark-Ewing Spaces and Groups

The Clark-Ewing construction of spaces with polynomial mod-p cohomology is closely related to standard ideas in group cohomology. In fact those spaces are simply cohomological torsion free analogues of classifying spaces for groups of the from (ℤ/p^{r}ℤ)^{×n} ⋊ W, where W < GL_{n}(ℤ/p^{r}ℤ) satisfies a certain condition. In this chapter we briefly recall the Clark-Ewing construction and discuss some elementary properties. Than we consider the associated finite groups and show that most of them satisfy Cohen’s conjecture.

The material presented in the first three sections is elementary in nature and is well known to the expert. It is presented here for the sake of completeness and for reference in section 6.5.

6.1. Construction

Let α : W ⟶ GL_{n}(ℤ_{p}^{∧}) be a faithful representation of a finite group W. The action of W on (ℤ_{p}^{∧})^{×n} via α induces an action on the Eilenberg-McLane space K(ℤ_{p}^{∧}, 2)^{×n}. Let X(W_{α}, p, n) denote the Borel construction EW ×_{W} K(ℤ_{p}^{∧}, 2)^{×n}, where EW is a free contractible right W -space and W acts on EW ×K(ℤ_{p}^{∧}, 2)^{×n} diagonally. One immediately observes that if W has order prime to p then the cohomology algebra H^{∗}(X(W_{α}, p, n); R) is the ring of invariants P [u_{1}, ⋯ u_{n}]^{W}, |u_{i}| = 2, 1 ≤ i ≤ n, for any p-local ring of coefficients R. The action of W on P [u_{1}, ⋯ u_{n}] is of course the one induced by the representation α.

Let R be a fixed commutative ring with a unit. Recall that an element a ∈ GL_{n}(R) is called a pseudo-reflection if it leaves a hyperplane in R^{n} fixed. A subgroup W < GL_{n}(R) is said to be a pseudo-reflection group if it is generated by pseudo-reflections in GL_{n}(R). A faithful representation α : W ⟶ GL_{n}(R) is said to be a pseudo-reflection representation if its image in GL_{n}(R) is a pseudoreflection group.

A classical result of [15, 56, 8] shows that if W is a finite group and R is a field of characteristic either zero or prime to the order of W, then a faithful representation α : W ⟶ GL_{n}(R) is a pseudo-reflection representation if and only if the ring of invariants P [t_{1}, ⋯ t_{n}]^{W} is again a polynomial ring. In [16] Clark and Ewing used this result together with a work of Shephard and Todd [56], on classification of irreducible complex pseudo-reflection representations, in order to write down a list of spaces of the form X(W, p, n), whose mod-p cohomology ring is a polynomial algebra. These spaces will be referred to as Clark-Ewing spaces. The Clark-Ewing list is given below. The reader is referred to [16] and [56] for details. A comprehensive but not as detailed treatment can be found in [34, chp. 22-23].

rank dimension of generators restrictions on p
1 n [4,6,⋯,2(n+1)] p > n + 1
2a^{∗} n [2m,4m,⋯,2(n-1)m,2qn] p ≡ 1 mod(m), p > n
2b  2  [4,2m]  p ≡ ±1 mod(m), m > 2
3  1  [2m]  p ≡ 1 mod(m )
4  2  [8,12]  p ≡ 1 mod(3 )
5  2  [12,24]  p ≡ 1 mod(3 )
6  2  [8,24]  p ≡ 1 mod(12 )
7  2  [24,24]  p ≡ 1 mod(12 )
8  2  [16,24]  p ≡ 1 mod(4 )
9  2  [16,48]  p ≡ 1 mod(8 )
10  2  [24,48]  p ≡ 1 mod(12 )
11  2  [48,48]  p ≡ 1 mod(24 )
12  2  [12,16]  p ≡ 1, 3 mod(8), p ≠ 3
13  2  [16,24]  p ≡ 1 mod(8 )
14  2  [12,48]  p ≡ 1, 19 mod(24 )
15  2  [24,48]  p ≡ 1 mod(24 )
16  2  [40,60]  p ≡ 1 mod(5 )
17  2  [40,120]  p ≡ 1 mod(20 )
18  2  [60,120]  p ≡ 1 mod(15 )
19  2  [120,120]  p ≡ 1 mod(60 )
20  2  [24,60]  p ≡ 1, 4 mod(15 )
21  2  [24,120]  p ≡ 1, 49 mod(60 )
22  2  [24,40]  p ≡ 1, 9 mod(20 )
23  3  [4,12,20]  p ≡ 1, 4 mod(5 )
24  3  [8,12,28]  p ≡ 1, 2, 4 mod(7)
25  3  [12,18,24]  p ≡ 1 mod(3 )
26  3  [12,24,36]  p ≡ 1 mod(3 )
27  3  [12,24,60]  p ≡ 1, 4 mod(15 )
28  4  [4,12,16,24]  p ≠ 2or3
29  4  [8,16,12,40]  p ≡ 1 mod(4), p ≠ 5
30  4  [4,24,40,60]  p ≡ 1, 4 mod(5 )
31  4  [16,24,40,48]  p ≡ 1 mod(4), p ≠ 5
32  4  [24,36,48,60]  p ≡ 1 mod(3 )
33  5  [8,12,20,24,36]  p ≡ 1 mod(3 )
rank dimension of generators restrictions on p
34 6  [12,24,36,48,60,84]  p ≡ 1 mod(3), p ≠ 7
35 6  [4,10,12,16,18,24]  p ≠ 2, 3 or 5
36 7  [4,12,16,20,24,28,36] p ≠ 2, 3, 5 or 7
37 8 [4,16,24,28,36,40,48,60] p ≠ 2, 3, 5 or 7

* In case 2a the integer m is divisible by q.

6.2. Spherical Resolutions of Loop Spaces on Clark-Ewing Spaces

Let X_{j}, j = 1, 2a, 2b, 3 ⋯ 37, denote the p-completion of the j-th Clark-Ewing space given in the table above. We use this symbol of course provided the corresponding space exists at the prime p. Also for the families 1, 2a, 2b and 3 we associate the corresponding parameters n and m, but write by abuse of notation X_{j} in those cases as well if no confusion can arise.

The existence of S-resolutions for most spaces of the form ΩX_{p}^{∧},where X is a Clark-Ewing space at the prime p follows as an immediate corollary from theorem 5.1.3 above and a the “big collapse theorem” of L. Smith [57], which allows an easy cohomology computation for those spaces.

Lemma 6.2.1. Let p be an odd prime. Let X be a simply connected space such that H^{∗}(X; 𝔽_{p}) ≅ P [x_{1}, x_{2}, ⋯ x_{n}]. Then there is an isomorphism of algebras

H^{∗}(ΩX; 𝔽_{p}) ≅ E[y_{1}, y_{2}, ⋯ y_{n}]

where |y_{j}| = |x_{j}| − 1.

Proof. This is immediate by the big collapse theorem [57]

The loop-spaces of the following p-completed Clark-Ewing spaces do not satisfy the low rank condition of theorem 5.1.3: X_{2b} for m = 4 at the prime 3, X_{28} at the prime 5, X_{35} at the prime 7 and the family X_{2a} for p ≡ 1 mod (m). Except for these, all ΩX_{j} do satisfy the hypotheses of 5.1.3.

Two of these exceptions are also S-resolvable and we thank the referee for pointing this out. Indeed ΩX_{2b} for m = 4 and p = 3 is S-resolvable by the discussion of section 6.3 below. The space ΩX_{35} at the prime 7 is S-resolvable by [60]. In addition the space ΩX_{28} at the prime 5 may appear as the 5-completion of the Lie group F_{4}, in which case it is S-resolvable by [49].

6.3. Resolutions by Cohomological Considerations

Let p be a prime and let X be a simply-connected p-local space such that

H^{∗}(X; 𝔽_{p}) ≅ E[x_{1}, x_{2}, ⋯, x_{n}]; |x_{i}| = 2m_{i} − 1.

We now observe that for most primes p, the existence of an S-resolution for X does not depend upon an H-space structure.

The results here are elementary. We include a brief discussion for the sake of completeness and mainly for reference in the following sections where spaces of this cohomological form are obtained without any guarantee of an H-space structure.

It is a standard fact that spaces X such that their integral cohomology is a finitely generated exterior algebra on odd dimensional generators are homotopy equivalent to products of spheres after localizing at a sufficiently large prime. The following lemma is an easy example, which one could prove by inspection of the corresponding cell structure as reflected in cohomology.

Lemma 6.3.1. Let p be an odd prime. Let Y be a simply connected space such that

H^{∗}(Y; 𝔽_{p}) ≅ E[x_{2n−1}, x_{2k−1}], k ≥ n

as an algebra. Suppose that k − n < p − 1. Then Y is homotopy equivalent to a product of spheres.

The following lemma applies to arbitrary ranks but the result is rather restrictive.

Lemma 6.3.2. Let X be a simply-connected space such that

H^{∗}(X; 𝔽_{p}) ≅ E[x_{1}, x_{2}, ⋯, x_{n}]; |x_{i}| = 2m_{i} − 1, m_{i} ≤ m_{i+1}

as an algebra. Let N = (∑_{i=1}^{n−1} m_{i}) − \frac{n−4}{2} and suppose that N ≤ p. Then X is S-resolvable.

Proof. Let X^{(k)} denote the k-skeleton of X. Then there exists a map X^{(2m_{n}−1)} ⟶^{g} S^{2m_{n}−1} which collapses all the cells in X^{(2m_{n}−1)} except for the one corresponding to x_{n} to a single point. One easily observes that under our assumptions there are no obstructions for extending g to a map g_{1} : X ⟶ S^{2m_{n}−1} which takes the generator ι_{2m_{n}−1} ∈ H_{2m_{n}−1}(S^{2m_{n}−1}; 𝔽_{p}) to x_{n}. An easy Serre spectral sequence argument shows that the homotopy fibre X_{1} of g_{1} is a simplyconnected space with

H^{∗}(X_{1}; 𝔽_{p}) ≅ E[x_{1}, x_{2}, ⋯, x_{n−1}].

An induction on n completes the proof.

Lemmas 6.3.1 and 6.3.2 show that any space X such that H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(ΩX_{j}; 𝔽_{p}) as an algebra for j = 1, n = 2 or j = 2a, n = 2 or j = 2b..22 is spherically resolvable with two possible exceptions. These are given by X_{8} at the prime 5 and X_{16} at the prime 11. It is rather elementary to observe that these two cases are spherically resolvable by inspecting that an obstruction to the existence of a map from X to the corresponding higher sphere does not exist. Thus we have shown

Proposition 6.3.1. Let X be a simply-connected space whose mod-p cohomology algebra is isomorphic to H^{∗}(X_{j}; 𝔽_{p}), j = 1, n = 2 or j = 2a, n = 2 or j = 2b..22, as an algebra. Then X is S-resolvable of length 2.

We next turn into studying the spaces corresponding to Clark-Ewing spaces of rank 3.

Proposition 6.3.2. Let X be a simply-connected space with H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(ΩX_{j}; 𝔽_{p}) as algebras for j = 23..27, where in each case p is a prime such that X_{j} is defined. Then X is spherically resolvable of length 3.

Proof. First consider the cases j = 23, 24 or 27. We prove the claim for j = 23. The other two work in a similar fashion. Let p be fixed. Let N be as in proposition 6.3.2. Then N < 11 ≤ p. Thus by proposition 6.3.2, there is a fibration

F ⟶ X ⟶ S^{19}.

Also by proposition 6.3.1, the 14-skeleton X^{(14)} ≃ S^{3}×S^{11}. Since the composite:

S^{3} × S^{11} ⟶^{≃} X^{14} ⟶ X ⟶ S^{19}

is null homotopic, there is a lifting l : S^{3} × S^{11} ⟶ F. A simple diagram chasing now shows that l induces an 𝔽_{p}-cohomology isomorphism and thus is a homotopy equivalence.

The remaining two cases, j = 25, 26, need a slightly different analysis. Consider a simply-connected space X with the same cohomology algebra as ΩX_{25}. The respective prime p has to satisfy p ≡ 1 mod 3. Thus the smallest possible value of p is 7. The space X is obtained from its 23-rd skeleton by consecutively attaching three cells of dimensions 28, 34, 40 and 51. Thus the obstructions to extending the natural collapsing map X^{(23)} ⟶ S^{23} to X are in π_{i}S^{23} for i = 27, 33, 39 and 50 respectively. Showing that those groups vanish at the prime 7 is an elementary exercise and so a map g : X ⟶ S^{23} which takes the generator ι_{23} to x_{23} exists. For primes p ≥ 7 it follows from 6.3.1 that the fibre of g is homotopy equivalent to a product of spheres. A similar analysis resolves spaces whose cohomology is isomorphic to that of ΩX_{26}.

It is likely that one can work out most spaces X corresponding to Clark-Ewing spaces of higher rank in a similar fashion independently of an H-space structure. We satisfy ourselves however with the list provided above.

6.4. Some Preliminaries from Representation Theory

Let p be a fixed prime and W be a finite group of order prime to p. We recall some useful facts on representations of W in the general linear groups over 𝔽_{p}, ℤ/p^{r}ℤ, and ℤ_{p}^{∧}. Let R and S be commutative rings with unit. Let f : R ⟶ S be a ring homomorphism. Then f induces a group homomorphism f̄ : GL_{n}(R) ⟶ GL_{n}(S). In particular let ρ : ℤ/p^{r}ℤ ⟶ ℤ/p^{r−1}ℤ be the reduction homomorphism and let ρ̄ be the induced group homomorphism. The following proposition and its corollary are proven in [34]

Proposition 6.4.1. Let W be a finite group of order prime to p. Let α_{r} : W ⟶ GL_{n}(ℤ/p^{r}ℤ) be a faithful representation. Then:

1. There exists a faithful representation α_{r+1} : W ⟶ GL_{n}(ℤ/p^{r+1}ℤ) such that the following diagram commutes:

[figure: W; =; GL_{n}(ℤ/p^{r+1}ℤ); ρ̄; α_{r+1}; W; GL_{n}(ℤ/p^{r}ℤ); α_{r}]

2. If α′_{r} and α′′_{r} are equivalent representations of W in GL_{m}(ℤ/p^{r}ℤ) and α′_{r+1}, α′′_{r+1} are lifts of α′_{r} and α′′_{r} respectively, then α′_{r+1} and α′′_{r+1} are also equivalent.

3. If α_{1} is a pseudo-reflection representation of a finite group W in GL_{n}(𝔽_{p}) and α_{r} is a lift of α_{1} to GL_{n}(ℤ/p^{r}ℤ) then α_{r} is a pseudo-reflection representation of W in GL_{n}(ℤ/p^{r}ℤ).

Since ℤ_{p}^{∧} ≅ lim_{\overleftarrow{r}} ℤ/p^{r}ℤ, we also have

Corollary 6.4.1. Let W be a finite group of order prime to p. Then equivalence classes of faithful representations of W in GL_{n}(ℤ/p^{r}ℤ) and in GL_{n}(ℤ_{p}^{∧}) are in a 1-1 correspondence. Furthermore if α_{r} is a pseudo-reflection representation of W in GL_{n}(ℤ/p^{r}ℤ) then any lift of α_{r} to GL_{n}(ℤ_{p}^{∧}) is a pseudo-reflection representation.

6.5. Clark-Ewing Groups

Throughout this section H_{∗}(−) and H^{∗}(−) will denote mod-p homology and cohomology respectively.

The spaces constructed by Clark and Ewing, or more generally, the spaces of the form X(W_{α}, p, n) are in some sense “torsion free analogues” of finite groups whose Sylow p-subgroup is isomorphic to (ℤ/p^{r}ℤ)^{×n}. In this section we discuss this analogy.

Definition 6.5.1. Let W be a finite group of order prime to p. Let α : W ⟶ GL_{n}(ℤ/p^{r}ℤ) be a faithful representation. Define T (W_{α}, p, n) to be the Borel construction EW ×_{W} (Bℤ/p^{r}ℤ)^{×n}, where W acts on (Bℤ/p^{r}ℤ)^{×n} via the representation α.

If X = X(W_{α}, p, n) is a Clark-Ewing space, then we shall relate to the group (ℤ/p^{r}ℤ)^{×n} ⋊_{α} W as the Clark-Ewing group of exponent p^{r} corresponding to X. Clearly T (W_{α}, p^{r}, n) ≃ B((ℤ/p^{r}ℤ)^{×n} ⋊_{α} W). The reason for using the Borel construction, rather than the classifying construction, is because the first behaves better with respect to construction of a certain map, which is given below.

Proposition 6.5.1. Let W be a finite group of order prime to p. Let a faithful representation α_{r} : W ⟶ GL_{n}(ℤ/p^{r}ℤ) be given. Let α̂ : W ⟶ GL_{n}(ℤ_{p}^{∧}) be a p-adic representation, which reduces to α_{r}. Then there exists a canonical map

γ_{r} : T (W_{α_{r}}, p^{r}, n) ⟶ X(Wα̂, p, n).

Furthermore, γ_{r}^{∗} is injective on mod-p cohomology and maps H^{∗}(X(Wα̂, p, n)) onto the subring P [u_{1} ⋯ u_{n}]^{W_{αr}} of H^{∗}(T (W_{α_{r}}, p^{r}, n)).

Proof. The natural map K(ℤ/p^{r}ℤ, 1)^{×n} ⟶ K(ℤ_{p}^{∧}, 2)^{×n} is evidently W - equivariant . Thus we obtain an induced map on Borel constructions

1 ×_{W} j : EW ×_{W} (Bℤ/p^{r}ℤ)^{×n} ⟶ EW ×_{W} K(ℤ_{p}^{∧}, 2)^{×n}.

Take γ_{r} = 1 ×_{W} j. The effect of γ_{r} on cohomology is obvious by construction and the proof is complete.

Proposition 6.5.1 is particularly useful when W admits a pseudo-reflection representation α : W ⟶ GL_{n}(𝔽_{p}), i.e. when X(Wα̂, p, n) is a Clark-Ewing space. In that case the spaces T (W_{α_{r}}, p^{r}, n), r ≥ 1 have a nice cohomological structure. The computation is due to J. Aguadé and L. Smith [3], who showed that if W is a finite group of order prime to p, which admits a pseudo-reflection representation α as above then

H^{∗}(T (W_{α}, p, n)) ≅ P [v_{1}, v_{2} ⋯, v_{n}] ⊗ E[z_{1}, z_{2}, ⋯, z_{n}],

with |v_{i}| = 2m_{i}, |z_{i}| = 2m_{i} − 1 and βz_{i} = v_{i}. Their proof easily generalizes to the computation of H^{∗}(T (W_{α_{r}}, p^{r}, n)).

Proposition 6.5.2 (Aguadé, Smith). Let α_{r} : W ⟶ GL_{n}(ℤ/p^{r}ℤ) be a pseudo-reflection representation of a finite group W of order prime to p. Then

H^{∗}(T (W_{α_{r}}, p^{r}, n)) ≅ P [v_{1}, v_{2} ⋯, v_{n}] ⊗ E[z_{1}, z_{2}, ⋯, z_{n}]

with |v_{i}| = 2m_{i}, |z_{i}| = 2m_{i} − 1 and β_{r}(z_{i}) = v_{i}, i = 1 ⋯ n.

Let W be a finite group of order prime to p. Suppose a pseudo-reflection representation α_{1} : W ⟶ GL_{n}(𝔽_{p}) is given. Let α_{r} : W ⟶ GL_{n}(ℤ/p^{r}ℤ) and α̂ : W ⟶ GL_{n}(ℤ_{p}^{∧}) be representations over ℤ/p^{r}ℤ and ℤ_{p}^{∧} respectively, which reduce to α_{1}. For every r ≥ 1 we have a map

γ_{r} : T (W_{α_{r}}, p^{r}, n)_{p}^{∧} ⟶ X(Wα̂, p, n)_{p}^{∧}

given by proposition 6.5.1. For a fixed group W and representation α_{1}, let F_{r} denote the homotopy fibre of γ_{r}.

Lemma 6.5.1. With the assumptions above, for every r ≥ 1,

H^{∗}(F_{r}) ≅ H^{∗}(ΩX(Wα̂, p, n)_{p}^{∧})

as algebras.

Proof. This is immediate from 6.5.1 and 6.5.2 together with the big collapse theorem of [57].

Let p be an odd prime and W be a finite group of order prime to p. Let a pseudo-reflection representation α_{1} : W ⟶ GL_{n}(𝔽_{p}) be given. Let α_{r} : W ⟶ GL_{n}(ℤ/p^{r}ℤ) and α̂ : W ⟶ GL_{n}(ℤ_{p}^{∧}) be faithful representations over ℤ/p^{r}ℤ and ℤ_{p}^{∧} respectively, which reduce to α_{1}. Let X denote X(Wα̂, p, n)_{p}^{∧}. Let T_{r} denote T (W_{α_{r}}, p^{r}, n)_{p}^{∧} and let F_{r} denote, as in proposition 6.5.1, the homotopy fibre of the natural map γ_{r} : T_{r} ⟶ X. The fibration

F_{r} ⟶ T_{r} ⟶ X

gives rise to a principal fibration

ΩF_{r} ⟶ ΩT_{r} ⟶ ΩX. (6.5.1)

Recall from section 6.2 that if X is a p-completed Clark-Ewing space then ΩX is S-resolvable. Some conditions under which F_{r} is S-resolvable were studied in section 6.3 above. Thus we get

Proposition 6.5.3. Suppose that X_{j} is a Clark-Ewing space for j = 2b..27. Let T_{r} be as above. Then ΩT_{r} is S-resolvable.

Proof. This follows by applying lemma 3.2.1 to the fibration 6.5.1 above and using the resolution constructed in section 6.3 for the respective F_{r}.

Remark 6.5.1. Proposition 6.5.3 is an example for the way in which Clark-Ewing groups could be handled. There are of course more examples one could work out in a similar fashion. Unfortunately we do not have a more conceptual method which could provide S-resolutions for ΩBπ_{p}^{∧} for every Clark-Ewing group.

Remark 6.5.2. If X is a Clark-Ewing space and T_{r} is the corresponding p^{r}- torsion space then, using the terminology of [21], the homotopy fibre F_{r} of γ_{r} : T_{r} ⟶ X is an H_{0}-space. Cooke, Harper and Zabrodsky provide in [21] a criterion for an H_{0}-space of low rank (< p − 1) to be an H-space, in which case it will also be S-resolvable. The question whether or not the low rank spaces of the form F_{r} are in fact H-spaces remains open.

CHAPTER 7 Discussion

We end this paper by discussing some further conjectures related to spaces of the form ΩBπ_{p}^{∧} for π finite and p-perfect.

As pointed out in the introduction to part 2, a counter example for the mod-p Cohen conjecture was recently discovered [37]. In view of the examples exhibited above , it might be interesting to ask what possible restrictions on a finite pperfect group π would imply that ΩBπ_{p}^{∧} is finitely S-resolvable. We point out that in all our examples the cohomology of the groups under consideration admits a filtration such that the associated graded has the form of a symmetric algebra. One might wonder whether this is a necessary condition for Cohen’s conjecture to hold.

We also point out that in all our examples the space ΩBπ_{p}^{∧} is resolved by spheres and single loop spaces on spheres, in contrast to the general conjecture which is stated in terms of iterated loop spaces on spheres. It might be interesting to look for an example where iterated loop spaces on spheres are essential for a resolution.

Finally the counter example for the mod-p Cohen conjecture given in [37] does not rule out the possibility that a suitable generalization of the conjecture might hold. For example it seems natural to include iterated loop spaces on Moore spaces in the family of resolving spaces. It might also be the case that spaces of the form ΩBπ_{p}^{∧} are S-resolvable but not of finite length. As for the last point, we remark that the question what are the spaces which are spherically resolvable of arbitrary length was already stated in [17]. However it is not at all clear that spaces of the form ΩBπ_{p}^{∧} have a better chance to be spherically resolvable than any quasibounded space (see part 1 section 6 for the definition of quasibounded).

Spaces of the form ΩBπ_{p}^{∧} tend to exhibit a rather interesting behavior in other aspects as well. In view of the existence of integral homology and stable homotopy exponents for ΩBπ_{p}^{∧}, one might wonder whether Bπ_{p}^{∧} admits an unstable homotopy exponent, possibly higher than the order of the Sylow p-subgroup (see theorem 1.1.1 of part 2). We point out that the existence of fibrations of the form

ΩBπ_{p}^{∧} ⟶ SU(n)_{p}^{∧} ⟶ (SU(n)/π)_{p}^{∧},

used in the proof of proposition 6.1.1 of part 1, for any finite p-perfect group π implies that if the Moore finite exponent conjecture [20] is true then Bπ_{p}^{∧} has a homotopy exponent. On the other hand in view of the fact that SU(n)_{p}^{∧} is spherically resolvable and thus has a homotopy exponent, an example of a finite p-perfect group π such that Bπ_{p}^{∧} does not have an exponent would provide a counter example to the Moore conjecture. Thus it seems reasonable to state

Conjecture 7.0.1. Let π be a finite p-perfect group. Then Bπ_{p}^{∧} admits an unstable homotopy exponent.

In all the examples known to the author the homology ring of ΩBπ_{p}^{∧} is a finitely generated algebra. It seems conceivable that this might hold in general. Thus we suggest the following

Conjecture 7.0.2. Let π be a finite p-perfect group. Then H_{∗}(ΩBπ_{p}^{∧}; 𝔽_{p}) is a finitely generated algebra.

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