The Homotopic Uniqueness of BS^{3}

William G. Dwyer Haynes R. Miller Clarence W. Wilkerson^{∗}

1 Introduction

Let p be a fixed prime number, F_{p} the field with p elements, and S^{3} the unit sphere in R^{4} considered as the multiplicative Lie group of norm 1 quaternions. The purpose of this paper is to prove the following theorem.

\underline{1.1 Theorem}. If X is any space with H^{∗}(X, F_{p}) isomorphic to H^{∗}(BS^{3}, F_{p}) as an algebra over the mod p Steenrod algebra, then the p-completion of X is homotopy equivalent to the p-completion of BS^{3}.

\underline{1.2 Remark}. It is easy to see that 1.1 implies that there is up to homotopy only one space B whose loop space is homotopy equivalent to the p-completion of S^{3}. To get such a strong uniqueness result it is definitely necessary to work one prime at a time: Rector [R] has produced an uncountable number of homotopically distinct spaces Y with loop space homotopy equivalent to S^{3} itself.

Rector’s deloopings {Y } have the property that Y_{p} ≃ (BS^{3})_{p}^{∧} for all primes p. Theorem 1.1 implies that this condition is forced. Thus Rector’s classification of the genus of BS^{3} actually classifies all deloopings of S^{3}. McGibbon [Mc 1] proved in 1978 that any delooping of S^{3} is stably equivalent at each odd prime to the standard BS^{3}. Rector [R] for odd primes and McGibbon [Mc 2] for p = 2 showed that the existence of a maximal torus in the sense of Rector distinguishes BS^{3} from other members of its genus, and hence by 1.1, from other deloopings of S^{3}.

∗The research of the authors was partially supported by the NSF, and that of the third author by sabbatical funds from Wayne State University.

\underline{1.3 Remark}. Theorem 1.1 is in some sense a delooping of the results of [M] and [D-M], although our techniques are somewhat different, especially in the complicated case p=2.

\underline{1.4 Organization of this paper}. Section 2 contains an account of the main background material we will need from [La] and [D-Z]. Section 3 treats the odd primary case of 1.1, and section 5 the case p=2. The intervening section 4 describes a new way of homotopically constructing BS^{3} at the prime 2. Section 6 is essentially an appendix which contains the proof of an auxiliary result needed in section 5.

\underline{1.5 Notation and terminology.} Some of the methods in this paper are based on simplicial techniques, so we will occasionally use “space” to mean “simplicial set” and tacitly assume that any topological space involved in the argument has been replaced by its singular complex [Ma]. In particular, X_{p}^{∧} (or X^{∧}, if p is understood) will denote the simplicial p-completion of the space X in the sense of [B-K,VII,5.1]. The space X is p-complete if the natural map X → X_{p}^{∧} is a weak equivalence. A p-completion X_{p}^{∧} is itself p-complete iff the map X → X_{p}^{∧} induces an isomorphism on mod p homology; this map does give an isomorphism, for instance, if H_{1}(X, F_{p}) = 0 or if X is connected and π_{1}(X) is a finite group [B-K]. Theorem 1.1 is equivalent to the claim that any p-complete space with the stated cohomology is homotopy equivalent to the p-completion of BS^{3}.

If X and Y are spaces, then Hom(X, Y) denotes the full function complex of maps X → Y; the subscripted variant Hom(X, Y)_{f} stands for the component of Hom(X, Y) containing a particular map f. As usual, [X, Y] denotes the set of components of Hom(X, Y), i.e., the set of homotopy classes of maps from X to Y.

If G is a (simplicial) group, then EG → BG is the functorial universal simplicial principal G−bundle [Ma,p83]. If G is abelian, then BG is also an abelian simplicial group and the classifying process can be iterated to form B^{2}G = B(BG), B^{3}G, ⋯, etc. The symbol σ will denote the cyclic group Z/pZ of order p and κ the additive group of p-adic integers, so that, for example, Bσ is equivalent to an infinite lens space and B^{2}κ to (CP^{∞})_{p}^{∧}.

Unless otherwise specified, all homology and cohomology is taken with simple F_{p} coefficients.

\underline{1.6 Remark.} The goal of this paper is to present the complete picture for BS^{3}. Obviously the present techniques carry over to other classifying spaces and exotic loop spaces. The discussion of section 3 applies almost verbatim to other rank 1 loop spaces, and with a few changes to the general case with p prime to the order of the Weyl group. On the other hand, the work of sections 4,5 and 6 for p = 2 is special to S^{3}, and must be replaced with new constructions for the small prime cases. As yet no general methods for these primes which divide the order of the Weyl group are available.

2 Background

Fix the prime p, and let K denote the category of unstable algebras over the mod p Steenrod algebra A_{p} [La]. For any two spaces X and Y there is a natural map

[Y, X] ⟶ Hom_{K}(H^{∗}(X), H^{∗}(Y))

\underline{2.1 Theorem} [La]. If X is a p-complete space with the property that H^{∗}(X) is finite in each dimension, then for each n ≥ 1 the map

[Bσ^{n}, X] ⟶ Hom_{K}(H^{∗}(X), H^{∗}(Bσ^{n}))

is an isomorphism.

Lannes has studied the functor T : K → K which is left adjoint to the functor given by tensor product with H^{∗}(Bσ^{n}). It is clear that the evaluation map

Hom(Bσ^{n}, X) × Bσ^{n} ⟶ X

induces for any space X a cohomology map

T^{n}H^{∗}(X) ⟶ H^{∗}(Hom(Bσ^{n}, X)).

\underline{2.2 Theorem} [La] If X is a p-complete space with the property that H^{∗}(X) is finite in each dimension, then the above map

T^{n}H^{∗}(X) ⟶ H^{∗}(Hom(Bσ^{n}, X))

is an isomorphism under either of the following two assumptions:

(1) T^{n}H^{∗}(X) is zero in dimension 1, or

(2) there is a p-complete space Z and a map

Z ⟶ Hom(Bσ^{n}, X)

such that the induced cohomology map

T^{n}H^{∗}(X) ⟶ H^{∗}(Z)

is an isomorphism.

We will need a slight refinement of 2.2, which can be proved by the same argument. If X is any space and g : H^{∗}(X) → H^{∗}(Bσ^{n}) ≅ (H^{∗}(Bσ^{n}))⊗ F_{p} is a map in K, let g̃ : T^{n}H^{∗}(X) → F_{p} be its adjoint. If R^{0} is the ring (T^{n}H^{∗}(X))^{0}, then g̃ amounts to a ring homomorphism R^{0} → F_{p} ; let (T^{n}H^{∗}(X))_{g} denote the tensor product (T^{n}H^{∗}(X)) ⊗_{R^{0}} F_{p}, where the action of R^{0} on F_{p} is via g̃. If f : Bσ^{n} → X is an actual map, we will also write (T^{n}H^{∗}(X))_{f} for (T^{n}H^{∗}(X))_{g} where g = f^{∗} : H^{∗}(X) → H^{∗}(Bσ^{n}) is the induced cohomology morphism. It is easy to argue by functoriality that the natural map

T^{n}H^{∗}(X) ⟶ H^{∗}(Hom(Bσ^{n}, X))

factors down to a map

(T^{n}H^{∗}(X))_{f} ⟶ H^{∗}(Hom(Bσ^{n}, X)_{f})

for each f : Bσ^{n} → X.

\underline{2.3 Theorem} [La] If X is a p-complete space with the property that H^{∗}(X) is finite in each dimension and f : Bσ^{n} → X is a map, then the map

(T^{n}H^{∗}(X))_{f} ⟶ H^{∗}(Hom(Bσ^{n}, X)_{f})

is an isomorphism under either of the following two assumptions:

(1) (T^{n}H^{∗}(X))_{f} vanishes in dimension 1, or (2) there is a connected p-complete space Z and a map

Z → Hom(Bσ^{n}, X)_{f}

such that the corresponding cohomology map

(T^{n}H^{∗}(X))_{f} ⟶ H^{∗}Z

is an isomorphism.

There is one situation in which the algebraic functors of 2.1 and 2.3 can be computed. If π is a finite group and G is a compact Lie group, let Rep(π, G) denote the set of G-conjugacy classes of homomorphisms π → G. Passing to classifying spaces gives a map

Rep(π, G) ⟶ [Bπ, BG]

If φ : π → G is a particular homomorphism and Z_{G}(φ) ⊆ G denotes the centralizer of the image of φ, then the obvious product map

Z_{G}(φ) × π ⟶^{inc.×φ} G

also passes to the classifying space level and induces a map

BZ_{G}(φ) ⟶ Hom(Bπ, BG)_{Bφ}

\underline{2.4 Theorem} [La, Ad, M-W] Suppose that G is a compact Lie group and that n ≥ 1. Then

(1) the natural composite

Rep(σ^{n}, G) → [Bσ^{n}, BG] → Hom_{K}(H^{∗}(BG), H^{∗}(Bσ^{n}))

is an isomorphism of sets, and

(2) for each homomorphism φ : σ^{n} → G the natural map

BZ_{G}(φ) → Hom(Bσ^{n}, BG)_{Bφ}

induces a composite map

T^{n}(H^{∗}(BG)_{Bφ}) ⟶ H^{∗}(Hom(Bσ^{n}, BG)_{Bφ}) ⟶ H^{∗}(BZ_{G}(φ))

which is an isomorphism in the category K.

The following lemma allows 2.1 - 2.3 to be applied in some cases to spaces which are not p-complete.

\underline{2.5 Proposition} If X is a connected space such that π_{1}(X) is a finite p-group, then for any finite p-group π the natural map

Hom(Bπ, X) ⟶ Hom(Bπ, X_{p}^{∧})

induces an isomorphism on mod p cohomology.

\underline{Proof} This is a consequence of [B-K,proof of VII,5.1] and the obstruction theory argument of [D-Z].

We will need to apply 2.5 in one case in which the group π involved is not elementary. The necessary key piece of information for the application comes from the following theorem.

\underline{Theorem} [D-Z]. If G is a compact Lie group and π is a finite p-group, then the natural map

Rep(π, G) ⟶ [Bπ, BG]

is an isomorphism of sets.

3 The odd primary case

In this section we will work under the assumption that p is a fixed odd prime and that X is a p-complete space with a fixed K-isomorphism H^{∗}(X) ≈ H^{∗}(BS^{3}). Let b_{2} denote a chosen generator of H^{2}(Bσ). If x is an element of H^{4}(X), then P^{2}(x) is either 2x^{\frac{p+1}{2}} or −2x^{\frac{p+1}{2}} ; let x_{4} ∈ H^{4}(X) denote a chosen generator of the first kind.

\underline{3.1 Proposition} There exists up to homotopy a unique map f : Bσ → X such that f^{∗}(x_{4}) = (b_{2})^{2}. The function space component Hom(Bσ, X)_{f} is homotopy equivalent to B^{2}κ.

\underline{Proof} It is clear that in the category K there is a unique map g : H^{∗}(X) ≈ H^{∗}(BS^{3}) → H^{∗}(Bσ) such that g(x_{4}) = (b_{2})^{2}; by 2.1, g is f^{∗} for a map f : Bσ → X which is unique up to homotopy. By 2.4, f^{∗} is the same as (Bφ)^{∗} for some (non-trivial) representation φ : σ → S^{3}. Since the centralizer of the image of φ in S^{3} is a circle group, it follows from 2.4 that (T H^{∗}(X))_{f} is isomorphic to H^{∗}(B^{2}κ). It follows from 2.3 that the same holds true of H^{∗}(Hom(Bσ, X)_{f}). The space Hom(Bσ, X)_{f} is p-complete in view of the fact that it is H_{∗}(−, F_{p})−local and has vanishing first homology group [Bo;12.6,proof of 4.3]], so the proof is finished by observing that B^{2}κ is determined up to homotopy among p-complete spaces by its cohomology ring.

For the rest of this section, we will fix a particular f which satisfies the condition of 3.1. Let e : Hom(Bσ, X)_{f} → X be the map obtained by evaluating a function at the basepoint of Bσ.

\underline{3.2 Proposition} The cohomology map

e^{∗} : H^{∗}(X) ⟶ H^{∗}(Hom(Bσ, X)_{f})

is injective.

\underline{Proof} Construct a commutative diagram

[figure: Bσ; Hom(Bσ, X)_{f}; X; r; f; e]

where the “right translation” map r is given by the formula (r(g))(a) = f(a + g). Here g, a ∈ Bσ and a + g denotes the sum of a and g with respect to the abelian group structure of Bσ (1.5). Commutativity follows from the fact that the basepoint of Bσ is the identity for the group structure. Since f^{∗} is injective, the proof is complete.

Let W = {1, w} be the group of order 2. Defining w : σ → σ to be the inverse map gives an action of W on σ which passes to a basepointpreserving action of W on Bσ and induces a right-composition action of W on Hom(Bσ, X).

\underline{3.3 Proposition} The above action of W on Hom(Bσ, X) carries the component Hom(Bσ, X)_{f} to itself.

\underline{Proof} It is necessary to check that f · w is homotopic to f, where

w : Bσ → Bσ

is given by the action of W. This follows from 3.1 and the fact that w^{∗} is multiplication by (-1) on H^{2}(Bσ), so that w^{∗}((b_{2})^{2}) = (−b_{2})^{2} = (b_{2})^{2}.

\underline{3.4 Proposition} The basepoint evaluation map e : Hom(Bσ, X)_{f} → X is equivariant with respect to the above action of W on Hom(Bσ, X)_{f} and the trivial action of W on X.

\underline{Proof} This is a formal consequence of the fact that the action of W on Bσ is basepoint-preserving.

\underline{3.5 Proposition} The automorphism w^{∗} of H^{2}(Hom(Bσ, X))_{f} induced by the action of W on Hom(Bσ, X)_{f} is multiplication by (-1).

\underline{Proof} Let r : Bσ → Hom(Bσ, X)_{f} be the map which appears in the proof of 3.2. Since w^{∗} : H^{2}(Bσ) → H^{2}(Bσ) is multiplication by (-1) and r^{∗} is injective (3.1, proof of 3.2), it is enough to show that the diagram

Bσ \underrightarrow{r} Hom(Bσ, X)_{f}
w ↓  ↓ w
Bσ \underrightarrow{r} Hom(Bσ, X)_{f}

commutes up to homotopy. Taking adjoints reduces this to the problem of showing that the two maps f · m · (1 × w) and f · m · (w × 1) are homotopic as maps Bσ × Bσ → X, where m : Bσ × Bσ → Bσ is the group multiplication. Since W acts on Bσ by group automorphisms, this follows from the fact that f · w is homotopic to f.

\underline{Proof of 1.1} (assuming p odd). Let Y denote the quotient space (Hom(Bσ, X)_{f} × EW)/W. By 3.4, the basepoint evaluation map e : Hom(Bσ, X)_{f} → X factors through a map ē: Y → X. By 3.1, 3.2, 3.5 and a calculation with the Serre spectral sequence, the map ē induces a mod p cohomology isomorphism H^{∗}(X) → H^{∗}(Y) and therefore a homotopy equivalence Y_{p}^{∧} → X_{p}^{∧} ∼ X. Since the order of W is prime to p and there is only one non-trivial homomorphism from W to the multiplicative group of p-adic integral units, it follows from 3.5 that up to homotopy the space Y is just the bundle over BW associated to the inversion action of W on B^{2}κ. In particular, Y does not depend on X, so the above proof that X is equivalent to Y_{p}^{∧} shows also that (BS^{3})_{p}^{∧} is equivalent to Y_{p}^{∧}.

4 Constructing BS^{3} at the prime 2

In this section we will assume that p is 2. The goal of this section is to find an explicit way of passing from some sort of finite group data to the 2-completion of BS^{3}.

Let Õ_{48} ⊆ S^{3} be the binary octahedral group, i.e. the inverse image in S^{3} ≈ Spin(3) of the group O_{24} in SO(3) of orientation-preserving isometries of the cube. The 2-Sylow subgroup of Õ_{48} is the quaternion group Q_{16} of order 16. Let NT ⊆ S^{3} be the normalizer of a maximal torus, and note that there is at least one injective homomorphism Q_{16} → NT.

\underline{4.1 Theorem} Assume that φ_{1} : Q_{16} → NT is an injective homomorphism and that φ_{2} : Q_{16} → Õ_{48} is the inclusion of a 2-Sylow subgroup. Let P be the homotopy pushout of the induced diagram

BÕ_{48} \underleftarrow{Bφ_{2}} BQ_{16} \underrightarrow{Bφ_{1}} BNT

Then the completion P_{2}^{∧} of P is homotopy equivalent to (BS^{3})_{2}^{∧}.

\underline{Proof} Let i : NT → S^{3} be the inclusion. Up to conjugacy the group Q_{16} has two faithful representations in S^{3} ≈ SU(2). These two representations have essentially the same image (since they differ by an outer automorphism of Q_{16} ) so we can assume that the composite iφ_{1} extends over φ_{2} to the representation Õ_{48} → S^{3} which was used above to define Õ_{48}. This gives a commutative diagram of classifying spaces

BQ_{16} \underrightarrow{Bφ_{1}} BNT
Bφ_{2} ↓  ↓ Bi
BÕ_{48} ⟶ BS^{3}

and leads to a map P → BS^{3}. To complete the proof it is enough to show that this map induces an isomorphism on mod 2 cohomology. Taking homotopy fibers over BS^{3} reduces the problem to showing that the homotopy pushout of the diagram

S^{3}/Õ_{48} ⟵ S^{3}/Q_{16} ⟶ S^{3}/NT

is mod 2 acyclic. This can be directly verified by a calculation with onedimensional integral cohomology, since S^{3}/Q_{16} and S^{3}/Õ_{48} are orientable 3manifolds while S^{3}/NT is RP^{2}.

Since the group S^{3} is a double cover of SO(3) , the following corollary is just a restatement of 4.1. Recall that the 2-Sylow subgroup of the octahedral group O_{24} is the dihedral group D_{8} of order 8, and that the normalizer of a maximal torus in S^{3} is isomorphic to O(2).

\underline{4.2 Corollary} Assume that φ_{1} : D_{8} → O(2) is an injective homomorphism and that φ_{2} : D_{8} → O_{24} is the inclusion of a 2-Sylow subgroup. Let P be the homotopy pushout of the induced diagram

BO_{24} \underleftarrow{Bφ_{2}} BD_{8} \underrightarrow{Bφ_{1}} BO(2)

then the completion P_{2}^{∧} is homotopy equivalent to (BSO(3))_{2}^{∧}.

\underline{4.3 Remark} The octahedral group O_{24} is isomorphic to the symmetric group on four letters, as well as to the semi-direct product constructed out of the natural action of the general linear group GL_{2}(F_{2}) on F_{2} × F_{2}.

5 The case of the prime 2

In this section we will assume that p is equal to 2 and that X is a 2-complete space with the property that H^{∗}(X) is isomorphic in the category K to H^{∗}(BS^{3}). (Note that there is a unique way to choose the isomorphism). Let x_{4} ∈ H^{4}(X) and b_{1} ∈ H^{1}(Bσ) label the generators of these groups.

\underline{5.1 Proposition} There exists up to homotopy a unique map f : Bσ → X such that f^{∗}(x_{4}) = (b_{1})^{4}. The basepoint evaluation map e : Hom(Bσ, X)_{f} → X is a homotopy equivalence.

\underline{Proof} By 2.1, there is a unique map f up to homotopy with the stated cohomological property and by 2.3 - 2.4 the cohomology ring H^{∗}(Hom(Bσ, X)_{f}) is isomorphic to H^{∗}(X). As in the proof of 3.1, then, we will be done if we can show that the cohomology map e^{∗} is bijective or even injective. Injectivity of e^{∗} follows as in the proof of 3.2 from the existence of a commutative diagram

[figure: Bσ; Hom(Bσ, X)_{f}; X; r; f; e]

Fix a map f as in 5.1. The right translation action of Bσ on itself (1.5) induces an action of Bσ on the function space component Hom(Bσ, X)_{f}. Let Y be the associated bundle over B^{2}σ, so that there is a principal fibration sequence

Bσ ⟶ X ⟶ Y.

\underline{5.2 Proposition} The space Y is 2-complete, and there is a unique K−isomorphism H^{∗}(Y) → H^{∗}(BSO(3)).

\underline{Proof} The fact that Y is 2-complete follows from [B-K]. By calculation, the E_{2}− term of the Rothenberg-Steenrod spectral sequence [R-S ]

E_{2}^{∗,∗} = Cotor_{H^{∗}Bσ}(H^{∗}(X), F_{2}) ⟹ H^{∗}(Y)

is a polynomial algebra on classes y_{2} and y_{3} in positions (1,1) and (1,2) respectively. For positional reasons, then, the spectral sequence collapses and y_{2}, y_{3} lift to unique classes (of the same name) in H^{∗}(Y) which generate H^{∗}(Y) as a polynomial algebra. The formula Sq^{1}y_{2} = y_{3} holds already at E_{2}, and it is \underline{a priori} clear that Sq^{2}y_{2} = (y_{2})^{2} and Sq^{3}y_{3} = (y_{3})^{2}. The class Sq^{2}y_{3} does not vanish, because Sq^{1}Sq^{2}y_{3} = Sq^{3}y_{3} = (y_{3})^{2} ≠ 0; inasmuch as there is only one non-zero cohomology class in H^{5}(Y), it follows that Sq^{2}y_{3} = y_{2}y_{3}. These formulas completely determine the action of the Steenrod algebra on H^{∗}(Y) and lead immediately to the desired result.

Let u_{1}, v_{1} ∈ H^{1}(Bσ^{2}) denote chosen generators.

\underline{5.3 Proposition} There exists up to homotopy a unique map g : Bσ^{2} → Y such that g^{∗}(y_{2}) = u_{1}^{2} + u_{1}v_{1} + v_{1}^{2} and g^{∗}(y_{3}) = u_{1}v_{1}(u_{1} + v_{1}). The map r : Bσ^{2} → Hom(Bσ^{2}, X)_{g} constructed as in the proof of 3.2 is a homotopy equivalence.

\underline{Proof} The existence and essential uniqueness of g follows from 2.1 and 2.4; g corresponds cohomologically to the unique faithful representation φ : σ^{2} → SO(3). To finish the proof, observe that for any space S and map h^{∗} : H^{∗}(S) → H^{∗}(Bσ^{2}), the Hopf algebra structure of H^{∗}(Bσ^{2}) provides a map T^{2}(H^{∗}(S))_{h^{∗}} → H^{∗}(Bσ^{2}). By 2.4 this map is an isomorphism if h = Bφ : Bσ^{2} → BSO(3); it follows that the map is also an isomorphism in the algebraically indistinguishable case h = g. The desired conclusion follows from 2.3, since the isomorphism T^{2}(H^{∗}(Y))_{g} → H^{∗}(Bσ^{2}) is derived from an actual map Bσ^{2} → Hom(Bσ^{2}, Y)_{g}.

Fix a map as in 5.3. Let W denote the general linear group GL_{2}(F_{2}); the group acts naturally on σ^{2} and therefore also on Bσ^{2} as well as on Hom(Bσ^{2}, Y). The following proposition is proved by the arguments of propositions 3.3 - 3.5.

5.4 Proposition

(1) The action of W on Hom(Bσ^{2}, Y) restricts to an action of W on Hom(Bσ^{2}, Y)_{g}.

(2) The basepoint evaluation map

e : Hom(Bσ^{2}, Y)_{g} ⟶ Y

is W-equivariant with respect to the trivial action of W on Y.

(3) The action of W on H_{1}(Hom(Bσ^{2}, Y)_{g}) is equivalent to the standard action of W on (F_{2})^{2}.

Proposition 5.4 immediately gives a map

ē: (EW × Hom(Bσ^{2}, Y)_{g})/W ⟶ Y.

A short calculation shows that the cohomology group H^{2}(GL_{2}(F_{2}), (F_{2})^{2}) vanishes if (F_{2})^{2} is given the standard GL_{2}(F_{2})−action; it follows, again by 5.4, that the fundamental group of the domain of ē is the octahedral group O_{24} (4.3). Recall that the 2-Sylow subgroup of O_{24} is the dihedral group D_{8} of order 8. Let s : BD_{8} → BO_{24} correspond to the inclusion of a 2-Sylow subgroup and c : Bσ → BD_{8} to the inclusion of the center.

\underline{5.5 Proposition} There exists a map

h : BO_{24} ⟶ Y

such that c^{∗}s^{∗}h^{∗}(y_{2}) = (b_{1})^{2}.

\underline{Proof} The map h is the map ē described above. The formula for the restriction of y_{2} to Bσ follows from 5.3 and the fact that, as in the proof of 3.2, h · s · c can be factored as a composite

Bσ ⟶ Bσ^{2} \underrightarrow{g} Y

in which the map Bσ → Bσ^{2} induces an inclusion σ → σ^{2}.

Let r : Bσ → Y denote the composite h · s · c above.

For the moment, use the map c above to identify σ with the center of D_{8}. The multiplication homomorphism D_{8} × σ → D_{8} gives rise to a composite

BD_{8} × Bσ ⟶ BD_{8} \underrightarrow{s} BO_{24} \underrightarrow{h} Y

The adjoint of this composite is a map

k : BD_{8} ⟶ Hom(Bσ, Y)_{r}

\underline{5.6 Proposition} The following diagram commutes:

BD_{8} \underrightarrow{k} Hom(Bσ, Y)_{r}
s ↓  ↓ e
BO_{24} h^{⟶} Y

Here, as usual, the right vertical map e is evaluation at the basepoint of Bσ.

\underline{5.7 Proposition} The function space component Hom(Bσ, Y)_{r} is homotopy equivalent to BO(2)_{2}^{∧}.

This will be proved in section 6.

\underline{Proof of 1.1} (for the prime 2). We will show that the space Y constructed above is homotopy equivalent to BSO(3)_{2}^{∧}, and leave it to the reader to deduce that X is equivalent to (BS^{3})_{2}^{∧}. Since c^{∗}k^{∗}e^{∗} = c^{∗}s^{∗}h^{∗} = r^{∗} ≠ 0 (5.5) the map k corresponds (2.4, 2.5) to an injective homomorphism D_{8} → O(2). Let P be the homotopy pushout of the diagram

BO_{24} \underleftarrow{s} BD_{8} \underrightarrow{k̃} BO(2)

in which k̃ is a lift (2.5) of k to BO(2). It follows from 4.2 that we will be done if we can show that the evident map l : P → Y induces an isomorphism on mod 2 cohomology. It is clear that l^{∗}(y_{2}) ≠ 0, since the restriction of y_{2} to the center of D_{8} is already non-zero; this implies that l^{∗} is an isomorphism in dimension 1. A calculation with Sq^{1} immediately shows that l^{∗} is also an isomorphism in dimension 2. The fact that both of the cohomology rings involved are polynomial algebras on generators of dimension 1 and 2 now completes the proof.

6 A function space calculation

In this section we will prove proposition 5.7. Notation and terminology will be taken from section 5. Recall from section 5 the cohomologically non-trivial map f : Bσ → X and the natural action of Bσ on Hom(Bσ, X)_{f}. It is easy to produce a similarly non-trivial map f′ : Bσ → B^{2}κ, as well as a corresponding action of Bσ on Hom(Bσ, B^{2}κ)_{f′}. Basepoint evaluation gives homotopy equivalences

\begin{matrix}Hom(Bσ, X)_{f} & ⟶ & X \\ Hom(Bσ, B^{2}κ)_{f′} & ⟶ & B^{2}κ \end{matrix}

Let Z denote the bundle over B^{2}σ associated to the diagonal action of Bσ on Hom(Bσ, X)_{f} × Hom(Bσ, B^{2}κ)_{f′}. Note that Z is just the analogue in this function space setting of the standard construction of U(2) as the quotient of SU(2) × S^{1} by Z/2Z.

\underline{6.1 Proposition} The space Z is 2-complete, and there is a (unique) K-isomorphism H^{∗}(Z) → H^{∗}(BU(2)).

\underline{Proof} The fact that Z is 2-complete follows from [B-K]. It is easy to argue that the bundle over B^{2}σ associated to the action of Bσ on Hom(Bσ, B^{2}κ)_{f′} has a total space homotopy equivalent to B^{2}κ itself. As a consequence, the Bσ−equivariant projection map

Hom(Bσ, X)_{f} × Hom(Bσ, B^{2}κ)_{f′} ⟶ Hom(Bσ, B^{2}κ)_{f′}

gives up to homotopy a fibration

X ⟶ Z ⟶ B^{2}κ.

The Serre spectral sequence of this fibration collapses and shows that H^{∗}(Z) is a polynomial algebra on classes z_{2} and z_{4} of dimensions 2 and 4 respectively. To show that H^{∗}(Z) is K-isomorphic to H^{∗}(BU(2)), it is necessary to check that Sq^{2}(z_{4}) is either z_{2}z_{4} or z_{2}z_{4} + (z_{2})^{2} (these two possibilities differ by the substitution of the alternative polynomial generator z_{4} + (z_{2})^{2} for z_{4}). This calculation can be made by examining the Eilenberg-Moore spectral sequence of the evident fibration

Z ⟶ Y ⟶ B^{3}κ.

This spectral sequence also collapses, and the residue class of z_{4} modulo (z_{2})^{2}is represented at E_{1} by the bar construction cycle z̃_{4} = y_{2} ⊗ i_{3} + 1 ⊗ Sq^{3}i_{3} (here i_{3} ∈ H^{3}(B^{3}κ) is the generator.) The fact that Sq^{1}i_{3} = 0 combines with the Cartan formula to give

Sq^{2}z̃_{4} = (y_{2})(z̃_{4}).

This formula implies the desired result, since the restriction of y_{2} to Z is z_{2}.

\underline{6.2 Proposition} The map r : Bσ → Y of 5.5-5.6 lifts to a map r̃: Bσ → Z which is unique up to homotopy. The space Hom(Bσ, Z)r̃ is homotopy equivalent to B^{2}κ^{2}.

\underline{Proof} This follows from 2.1, 2.3 and 2.4. The map r̃ corresponds cohomologically to the map Bσ → BU(2) derived from the regular representation of σ.

\underline{6.3 Proposition} The loop space ΩHom(Bσ, Y)_{r} is homotopy equivalent to the space σ × Bκ.

\underline{Proof} The fibration

Z ⟶ Y ⟶ B^{3}κ

gives rise to a function space fibration

Hom(Bσ, Z)r̃ ⟶ Hom(Bσ, Y)_{r} ⟶ Hom(Bσ, B^{3}κ)

(the fiber is as described because the map r : Bσ → Z composes to a nullhomotopic map Bσ → B^{3}κ). The homotopy groups of Hom(Bσ, B^{3}κ) are given by the table

π_{i}Hom(Bσ, B^{3}κ) = \{ \begin{matrix}σ & i = 1 \\ κ & i = 3 \\ ∗ & otherwise \end{matrix}

so the proof can be completed by showing that the boundary homomorphism

κ ≈ π_{3}Hom(Bσ, B^{3}κ) ⟶ π_{2}Hom(Bσ, Z)r̃ ≈ κ^{2}

remains non-zero after tensoring with F_{2}. Consider the diagram

σ × B^{2}κ ∼ Hom(Bσ, B^{2}κ) \underrightarrow{a} Hom(Bσ, Z)r̃ ∼ B^{2}κ^{2}
e_{1} ↓  ↓ e_{2}
B^{2}κ  \underrightarrow{b}  Z

in which the vertical arrows are obtained by evaluation at the basepoint of Bσ and the horizontal arrows from the fiber inclusion in the shifted fibration

(*) B^{2}κ \underrightarrow{b} Z ⟶ Y

The map a induces a morphism on π_{2} which is essentially the boundary homomorphism of interest, and the map e_{1} restricts to a homotopy equivalence from either component of σ × B^{2}κ to B^{2}κ. A look at the Serre spectral sequence of (*) shows that b^{∗} : H^{4}(Z) → H^{4}(B^{2}κ) is surjective, which implies that a^{∗} induces epimorphisms H^{4}(B^{2}k^{2}) → H^{4}({ε}×B^{2}κ) for each ε in σ. This directly implies the desired result.

\underline{Proof of 5.7} By 6.3, the space Hom(Bσ, Y)_{r} is equivalent to the 2-completion of one of the following three spaces

\begin{matrix}a) & Bσ × BS^{1} \\ b) & BO(2) \\ c) & BNT \end{matrix}

where NT ⊆ S^{3} is the normalizer of a torus. Possibilities (a) and (c) are ruled out by combining 5.6, 2.1, 2.4 and 2.5, since there is no homomorphism D_{8} → σ × S^{1} or D_{8} → NT which is non-trivial on the center of D_{8}.

References

[Ad] J.F. Adams, private communication, Fall, 1985.

[Bo] A.K. Bousfield, The localization of spaces with respect to homology, Topology, vol 14(1975), pp 133-150.

[B-K] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completion and Localizations, Lecture Notes in Math. 304, Springer (1972).

[D-M] W.G. Dwyer and G. Mislin, On the homotopy type of the components of Map_{∗}(BS^{3}, BS^{3}), to appear.

[D-Z] W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Proceedings of the 1986 Barcelona Conference, Springer Lecture Notes, to appear.

[La] J. Lannes, Cohomology of groups and function spaces, manuscript.

[Ma] J.P. May, \underline{Simplicial Objects in Algebraic Topology}, Van Nostrand (1976).

[Mc 1] Charles A. McGibbon, Which group structures on S^{3} have a maximal torus?, \underline{Geometric Applications of Homotopy} \underline{Theory I}, Springer Lecture Notes ♯657, pp 353-360.

[Mc 2] Charles A. McGibbon, Stable properties of rank 1 loop structures, Topology, v.20(2) 1981 pp 109-118.

[M] G. Mislin, The homotopy classification of self-maps of infinite quaternionic projective space, Quart. J. Math. Oxford, to appear.

[M-W] H. Miller and C. Wilkerson, Maps of elementary p-groups into compact Lie groups, lecture by second author at 1985 Yale Conference on Algebraic Topology.

[R] D. Rector, Loop structure on the homotopy type of S^{3}, Symposium on Algebraic Topology, (Batelle Seattle Research Center, Seattle, Wash. 1971) pp 99-105 Lecture Notes in Math. Vol 249, Springer (1971).

[R-S] M. Rothenberg and N.E. Steenrod, The cohomology of classifying spaces of H-spaces, Bull. Amer. Math. Soc. 71(1965), pp 872-875.