A New Finite Loop Space at the Prime Two

W. G. Dwyer and C. W. Wilkerson

University of Notre Dame Purdue University

Abstract. We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the first example of an exotic finite loop space at 2. We conjecture that it is also the last one.

§1. Introduction

From the point of view of homotopy theory a compact Lie group G has the following remarkable combination of properties:

(1) G can be given the structure of a finite CW-complex, and

(2) there is a pointed space BG and a homotopy equivalence from G

to the loop space ΩBG. Of course the space BG in (2) is the ordinary classifying space of G. In general, a finite complex X together with a chosen equivalence X → ΩBX for some BX is called a finite loop space. If p is a prime number and the geometric finiteness condition on X is replaced by the requirement that X be F_{p}-complete in the sense of [3] and have finite mod p cohomology, then X is called a p-adic finite loop space or a finite loop space at the prime p. A (p-adic) finite loop space is a strong homotopytheoretic analogue of a compact Lie group. The study of these spaces is related to many classical questions in topology (for instance, to the problem of determining all spaces with polynomial cohomology rings).

Call a p-adic finite loop space X exotic if it is not the F_{p}-completion of G for a compact Lie group G. There are many known examples of exotic p-adic finite loop spaces at odd primes p [5] and the classification of these spaces is partially understood [8] [9]. However, until now there have been no known exotic 2-adic finite loop spaces.

Recall [32] that the ring of rank 4 mod 2 Dickson invariants is the ring of invariants of the natural action of GL(4, F_{2}) on the rank 4 polynomial algebra H^{∗}((BZ/2)^{4}, F_{2}); this ring of invariants is a polynomial algebra on classes c_{8}, c_{12}, c_{14}, and c_{15} with Sq^{4}c_{8} = c_{12}, Sq^{2}c_{12} = c_{14}, and Sq^{1}c_{14} = c_{15}. Our main theorem is the following one.

This work was supported in part by the National Science Foundation.

1.1 Theorem. There exists an F_{2}-complete space BDI(4) such that H^{∗}(BDI(4), F_{2}) is isomorphic as an algebra over the Steenrod algebra to the ring of rank 4 mod 2 Dickson invariants.

Let DI(4) = ΩBDI(4). Standard methods show that H^{∗}(DI(4), F_{2}) is multiplicatively generated by elements x_{7}, y_{11}, and z_{13}, with Sq^{4}x = y, Sq^{2}y = z, Sq^{1}z = x^{2} ≠ 0, and x^{4} = y^{2} = z^{2} = 0. This space DI(4) is an exotic 2-adic finite loop space.

It is natural to ask about the realizability of other Dickson invariant algebras. Say that a space Y is of type BDI(n) if H^{∗}(Y, F_{2}) is isomorphic, as an algebra over the Steenrod algebra, to the algebra of rank n mod 2 Dickson invariants [32]. Then RP^{∞} = BZ/2 is of type BDI(1), B SO(3) is of type BDI(2) and the classifying space B G_{2} of the exceptional Lie group G_{2} is of type BDI(3). It is known that no space of type BDI(n) can exist for n ≥ 6 [31]. Lannes has recently used methods similar to ours to show the non-existence of a space of type BDI(5), while [24] proves the stronger statement that an H-space “of type DI(5)” does not exist (see also [18]). The construction of BDI(4) in this paper completes the determination of the set of integers n for which a space of type BDI(n) exists.

One indication that BDI(4) might exist comes from Lie theory. If G is a connected compact Lie group of rank r, then the Weyl group W_{G} is a finite subgroup of GL(r, Q) generated by reflections, and the rational cohomology ring of BG is naturally isomorphic to the ring of polynomial invariants of W_{G}. Finite reflection groups are relatively uncommon, and in fact G is close to being determined by its Weyl group W_{G}. Let Ẑ_{2} denote the ring of 2-adic integers. About ten years ago the second author observed that there exists a finite reflection subgroup W_{DI(4)} of GL(3, Q ⊗ Ẑ_{2}) such that the ring Q ⊗ H^{∗}(BDI(4), Ẑ_{2}) (which is easily seen to be a polynomial algebra on generators of dimensions 8, 12, and 28) is isomorphic to the ring of polynomial invariants of W_{DI(4)}. In a sense this group W_{DI(4)} is a plausible “Weyl group” for a 2–adic finite loop space of type DI(4). The group W_{DI(4)} is isomorphic to Z/2 × GL(3, F_{2}) and arises as complex reflection group (number 24 on the list of [5]) whose reflection representation can be defined over a subfield of C which embeds in Q ⊗ Ẑ_{2} [5, p. 431]. In 1980 there was no evident way to use the existence of W_{DI(4)} to construct BDI(4); even now we use the existence of W_{DI(4)} only in an indirect way (§4) and make no explicit reference to the reflection properties of the group. On the whole, though, the results of this paper suggest that there is a very close link between p-adic finite loop spaces and p-adic reflection groups. Recent work of the authors has carried this idea further by showing that each padic finite loop space determines a p-adic reflection group. On the basis of the classification of 2-adic reflection groups ([5], [9]) we are therefore led to conjecture that the classifying space of any connected 2-adic finite loop space is equivalent to the product of a number of copies of BDI(4) with the F_{2}-completion of the classifying space of a connected compact Lie group. 1.2 The basic technique: Our present method of building BDI(4) depends on the ideas of [20], [17] and [13]. The starting point is again the theory of Lie groups, but this time on the homotopy theoretic side. Let T denote the category of topological spaces. Jackowski and McClure have proved the following remarkable result:

1.3 Proposition. [17] For any compact Lie group G there exist

(1) a category A_{G}^{Lie} whose objects are the (conjugacy classes of) nontrivial elementary abelian 2-subgroups of G,

(2) a functor α_{G}^{Lie} : (A_{G}^{Lie})^{op} → T which up to homotopy assigns to each such subgroup the classifying space of its centralizer in G, and

(3) a map ho\underrightarrow{lim} α_{G}^{Lie} → BG which is an isomorphism on F_{2} cohomology.

Unless G has a central element of order 2, this proposition provides a decomposition of BG up to F_{2} cohomology as a generalized pushout of classifying spaces of proper subgroups of G. There is also an algebraic version of the above. Let K denote the category of unstable algebras over the mod 2 Steenrod algebra.

1.4 Proposition. [13, §2] For any object R of K there exist

(1) a category A_{R}^{alg} in which an object is a pair (V, f^{∗}) consisting of a non-trivial elementary abelian 2-group V together with (1.5) a suitable K-map f^{∗} : R → H^{∗}(BV, F_{2}),

(2) a Lannes functor α_{R}^{alg} : (A_{Y}^{alg})^{op} → K which sends the pair (V, f^{∗}) to an algebra T_{f^{∗}}^{V}(R) (see 1.5), and

(3) a map R → \underleftarrow{lim} α_{R}^{alg} which in favorable cases is an isomorphism.

Moreover if R = H^{∗}(BG, F_{2}) (G compact Lie) then [21] [13] the categories A_{R}^{alg} and A_{G}^{Lie} are equivalent in such a way that the functors α_{R}^{alg} and H^{∗}(α_{G}^{Lie}, F_{2}) correspond.

Now let Y be BDI(4) and let R = H^{∗}(Y, F_{2}) be the rank 4 Dickson invariant algebra. It is easy to figure out the category A_{R}^{alg} (see 1.5) and for each object (V, f^{∗}) of A_{R}^{alg} to compute the algebra α_{R}^{alg}(V, f^{∗}). It is natural to hope that Y is sufficiently similar to the classifying space of a compact Lie group that something like the Jackowski–McClure decomposition might work for Y. With this as motivation, then, we construct by hand (§7) a functor F : (A_{R}^{alg})^{op} → T such that H^{∗}(F, F_{2}) is naturally equivalent to α_{R}^{alg}, and verify by direct algebraic calculation (2.4) that the cohomology of the homotopy direct limit of this functor is the ring of rank 4 mod 2 Dickson invariants. We obtain BDI(4) as the F_{2}-completion of this homotopy direct limit.

To build the functor F we first (§6) construct a parallel functor F̄ with values in the homotopy category and then use the obstruction theory of [6] to lift F̄ to a functor F with values in the category of spaces. We are fortunate that this lifting problem is easy to solve once it is set up: all of the obstruction groups are zero (7.6).

1.5 The role of Spin(7): The construction of the above functor F̄ involves careful study of the Lie group Spin(7) and some of its subgroups. (Here Spin(7) is the connected double covering group of the special orthogonal group SO(7).) There is a simple explanation for this. Let R = H^{∗}(BDI(4), F_{2}) be the rank 4 Dickson invariant algebra. Because of 1.3 and 1.4 it is useful to refer to an object A = (V, f^{∗}) ∈ A_{R}^{alg} with rank(V) = r as a rank r elementary abelian 2–subgroup of DI(4) (taken up to conjugacy), and to interpret α_{R}^{alg}(A) as the cohomology of the classifying space of the centralizer of A in DI(4). Explicitly, then, the elementary abelian 2-subgroups of DI(4) correspond to pairs (V, f^{∗}) where V is an elementary abelian 2-group and f^{∗} : R → H^{∗}(BV, F_{2}) is a K-map which makes H^{∗}(BV, F_{2}) into a finitely generated module over R [13, §2]. The following lemma (due to Lannes) explains how to enumerate these and calculate the values of the functor α_{R}^{alg}.

We will let T^{V} denote the T -functor of Lannes [19]; T^{V} is the left adjoint on the category of unstable modules (or algebras) over the Steenrod algebra to the functor given by tensoring with H^{∗}(BV, F_{2}). If X is a space the algebra T^{V} (H^{∗}(X, F_{2})) is closely related to the F_{2} cohomology algebra of the space of maps from BV to X. Recall [11] that to any object S ∈ K and map f^{∗} : S → H^{∗}(BV, F_{2}) there is associated a quotient object T_{f^{∗}}^{V}(S) of T^{V} (S); if S = H^{∗}(X, F_{2}) then T_{f^{∗}}^{V}(S) is closely related to the F_{2} cohomology ring of the space of maps BV → X inducing f^{∗} on cohomology. (Note that in [13] the algebra T_{f^{∗}}^{V}(S) is denoted T (V, S)_{f}.)

1.6 Lemma. Let E be an elementary abelian 2-group, Γ a subgroup of Aut(E), and S the unstable algebra over the Steenrod algebra given by the fixed point set H^{∗}(BE, F_{2})^{Γ}. Let V be an elementary abelian 2-group.

(1) The map which assigns to each homomorphism f : V → E the

induced composite S ⟶ H^{∗}(BE, F_{2}) ⟶^{f^{∗}} H^{∗}(BV, F_{2}) gives a bijection between K-maps f^{∗} : S ⟶ H^{∗}(BV, F_{2}) and Γ-orbits of homomorphisms f : V → E.

(2) A map f^{∗} : S ⟶ H^{∗}(BV, F_{2}) makes H^{∗}(BV, F_{2}) into a finitely generated S module iff the corresponding (1) homomorphism f : V → E is a monomorphism.

(3) If f^{∗} : S ⟶ H^{∗}(BV, F_{2}) corresponds to f : V → E, then T_{f^{∗}}^{V}(S) is naturally isomorphic to H^{∗}(BE, F_{2})^{Γ_{f}}, where Γ_{f} ⊂ Γ is the subgroup of elements which pointwise fix the image of f.

Proof: By exactness of the functor T^{V} [21] there is an isomorphism

T^{V} (S) = T^{V} (H^{∗}(BE, F_{2})^{Γ}) ≅ (T^{V} (H^{∗}(BE, F_{2}))^{Γ} .

Moreover, the algebra T^{V} (H^{∗}(BE, F_{2})) is isomorphic [21, 3.4.5] as a Γ-object to the tensor product H^{0}(Hom(V, E), F_{2}) ⊗ H^{∗}(BE, F_{2}) with the diagonal Γ action (the first factor here is the zero-dimensional cohomology of the discrete space Hom(V, E); the tensor product itself in fact gives the cohomology of the space of maps BV → BE). Parts (1) and (3) follow from the definition of T_{f^{∗}}^{V} and inspection. Part (2) can be obtained for instance from [13, proof of 4.4].

Since the group Γ = GL(4, F_{2}) acts transitively on the subspaces of (Z/2)^{4} of any given dimension, it follows that up to conjugacy there is only one rank i elementary abelian 2–subgroup A_{i} of DI(4) for each i = 1, …, 4. Moreover, the algebra α_{R}^{alg}(A_{i}) is isomorphic to the fixed point set of the action of Γ_{i} on H^{∗}((BZ/2)^{4}, F_{2}), where Γ_{i} ⊂ GL(4, F_{2}) is the set of matrices which agree with the identity in the first i columns. A study of morphisms [13, §2] now implies that the category A_{R}^{alg} is equivalent to the category A of §2 and that the functor α_{R}^{alg} is equivalent to the functor α_{R} of 2.3. The category A has four objects and the following shape

A_{1} ⇒  A_{2}  ⇒  A_{3}  ⇒  A_{4}
|  |  |  |
{1}  GL(2, F_{2})  GL(3, F_{2})  GL(4, F_{2})

where under each object is its monoid (in this case group) of self maps and “⇒” stands for an appropriate set of morphisms. There are no morphisms A_{i} ⟶ A_{j} for i > j and for i ≤ j the group GL(j, F_{2}) = Aut(A_{j}) acts transitively on the set of morphisms A_{i} ⟶ A_{j}.

This calculation, though, reveals something perhaps unexpected: the cohomology ring α_{R}^{alg}(A_{1}) of the classifying space of the centralizer of the essentially unique rank 1 elementary abelian 2-subgroup of DI(4) (the cohomology ring calculated as H^{∗}((BZ/2)^{4}, F_{2})^{Γ_{1}} above) is exactly (3.11) the F_{2} cohomology ring of B Spin(7)! Encouraged by this, we surmise that this centralizer is Spin(7) and begin the construction of F̄ by declaring F̄(A_{1}) to be the F_{2}-completion of B Spin(7). It then stands to reason that the centralizer in DI(4) of a rank 2 elementary abelian 2-subgroup A, say, should be the same as the centralizer of A inside the centralizer within DI(4) of any rank 1 subgroup of A, i.e. the same as the centralizer of A within Spin(7). This can be verified cohomologically by combining 1.3, 1.4 and 1.6; verification comes down to observing that the fixed point set of the action of the group Γ_{i} (2 ≤ i ≤ 4) on H^{∗}((BZ/2)^{4}, F_{2}) doesn’t depend on whether Γ_{i} is encountered as a subgroup of Γ_{1} or as a subgroup of GL(4, F_{2}). The conclusion is (3.13) that for i = 2, 3, 4 the algebra α_{R}^{alg}(A_{i}) is isomorphic to H^{∗}(BC(E_{i}), F_{2}), where C(E_{i}) is the centralizer in Spin(7) of a rank i elementary abelian 2-group E_{i} containing the center of Spin(7). This is evidence enough for us to declare F̄(A_{i}) to be the F_{2}-completion of BC(E_{i}).

1.7 Final ingredients: Once we have settled on the spaces which the functor F̄ will take on as its values, we have to look for homotopy classes of maps between these spaces satisfying the relations described in 2.1. The basic maps F̄(A_{i}) → F̄(A_{j}) (i > j) arise from the inclusion of one subgroup of Spin(7) in another. No non-identity self-maps of F̄(A_{1}) are required, and the necessary self-maps of F̄(A_{4}) are easy to construct, since this space is just (BZ/2)^{4}. For i = 2 and i = 3, though, it turns out that inner automorphisms of Spin(7) provide actions of a certain matrix group (§2) P (1, i − 1) ⊂ GL(i, F_{2}) on F̄(A_{i}), whereas the existence of F̄ as a functor requires an action of the full group GL(i, F_{2}) on this space. The case i = 2 of this problem is solved in §5, and the case i = 3 in §4 (this last case is particularly interesting because it involves constructing what is essentially a 2-adic integral form of the basic reflection representation of W_{DI(4)}). A significant amount of effort goes into being explicit enough about these solutions that it is possible to verify concretely that the necessary diagrams commute up to homotopy and that they have the correct cohomological properties (see §6).

In the end our diagram looks like this:

B̂ Spin(7) ⇐ B̂(SU(2)^{3}/{±1}) ⇐ B̂((S^{1})^{3}×| Z/2) ⇐ B(Z/2)^{4}
|  |  |  |
{1}  GL(2, F_{2})  GL(3, F_{2})  GL(4, F_{2})

where “B̂X” denotes the F_{2} completion of “BX”. (See §3 for detailed descriptions of the above groups.) Of course the spaces in the above diagram exist integrally, but the maps we use exist only after F_{2} completion.

Finally, we note that the conflict between Theorem 1.1 and the nonrealizability results of [25] has been resolved in [23]. Lie-like properties: The construction of BDI(4) given in this paper begins with the hypothesis that this space has properties similar to those of the classifying space of a compact Lie group. Theorem 1.8, whose proof is computational but routine, points out a few striking Lie-like properties that follow from the construction

1.8 Theorem. The space BDI(4) has the following properties:

(1) There is a map f : B Spin(7) → BDI(4) such that the homotopy fibre DI(4)/ Spin(7) has F_{2} cohomology which is finite, concentrated in even degrees, and has Euler characteristic 24.

Let BT → B Spin(7) be induced by the inclusion of the (rank 3) maximal torus, let g denote the composite BT → B Spin(7) → BDI(4) and let DI(4)/T denote the homotopy fibre of g.

(2) H^{∗}(DI(4)/T, F_{2}) is finite, concentrated in even degrees, and has Euler characteristic 336 = |W_{DI(4)}|.

(3) The space of self-equivalences of B̂T (see 1.9) over BDI(4) is homotopically discrete and has W_{DI(4)} as its group of components (cf. [9, 2.8]).

The space DI(4)/ Spin(7) is perhaps an exotic symmetric space at the prime 2. The space DI(4)/T is the flag manifold (at the prime 2) corresponding to DI(4). 1.9 Notation and terminology: We will follow the standard convention that composition of maps between sets, groups, etc. works from right to left: a composite function f · g or fg is defined by the formula f · g(x) = f(g(x)). In particular, if X is a set, group, etc. the monoid Hom(X, X) acts in a natural way on X from the left.

Unless otherwise specified, cohomology is taken with coefficients in the field F_{2} of two elements. If X and Y are spaces then Map(X, Y) denotes the function space of maps from X to Y; the component containing a particular map or homotopy class f is Map(X, Y)_{f}. The space Emb(X, Y) is the subspace of Map(X, Y) containing maps with respect to which H^{∗}X is a finitely generated module over H^{∗}Y; Aut(X) is the topological monoid of self-homotopy equivalences of X with identity component Aut(X)_{1}. The symbol B̂G will stand for the F_{2}-completion of the classifying space BG.

The center of a (Lie) group G is denoted Z(G); if G is given as a subgroup of some ambient group H then C(G) and N (G) denote respectively the centralizer and normalizer of G in H. For general compact Lie groups G and H we will let Emb(G, H) stand for the set of H-conjugacy classes of group monomorphisms G → H; note [29, 2.4] that there are natural maps Emb(G, H) → π_{0}Emb(BG, BH) → π_{0}Emb(B̂G, B̂H).

The symbol Σ_{n} will stand for the symmetric group of degree n, i.e., the group of automorphisms of the set {1, …, n}. Suppose that G is a (topological) group or monoid. The wreath product Σ_{n} ≀ G is the space Σ_{n} × G^{n} with multiplication determined by the formula

[g_{1}, …, g_{n}] · σ = σ · [g_{σ(1)}, …, g_{σ(n)}] ,

where σ ∈ Σ_{n} and [g_{1}, …, g_{n}] belongs to the product monoid G^{n}.

The authors would like to thank D. Benson, R. Oliver, R. Kane and the referee for very helpful comments.

§2. The basic category

In this section we describe the basic index category A which is used in the paper; then we construct and study the algebraic diagram over A which we intend eventually to realize geometrically.

Let A be the category whose objects are the F_{2} vector spaces (Z/2)^{i}, i = 1, …, 4 and whose morphisms are vector space monomorphisms. Give the object (Z/2)^{i} (denoted A_{i}) the standard ordered basis e_{i}(j), j = 1, …, i, where e_{i}(j) is zero except in the j’th coordinate. It is clear that Hom_{A}(A_{i}, A_{j}) is empty if i > j, and that (in view of the chosen basis for A_{i}) the group Hom_{A}(A_{i}, A_{i}) is GL(i, F_{2}). We will think of the elements of A_{i} as column vectors, so that as usual (1.9) the group GL(i, F_{2}) acts on the elements of A_{i} from the left.

There are standard embeddings f_{i} : A_{i} → A_{i+1}, i = 1, …, 3 in A determined by the formula f_{i}(e_{i}(j)) = e_{i+1}(j). Let P (i, j) denote the subgroup of GL(i + j, F_{2}) consisting of matrices of the form

( \begin{matrix}M_{11} & M_{12} \\ 0 & M_{22} \end{matrix} )

where M_{11} ∈ GL(i, F_{2}), and let r_{i} : P (i, j) → GL(i, F_{2}) be the homomorphism which sends the above element of P (i, j) to the matrix M_{11}. The group P (i, j) is the subgroup of GL(i + j, F_{2}) given by transformations which carry to itself the subspace of A_{i+j} generated by the first i standard basis elements.

The following lemma is an elementary calculation.

2.1 Lemma. The category A is generated by the groups GL(i, F_{2}) = Hom_{A}(A_{i}, A_{i}), i = 1, …, 4 and the maps f_{i} : A_{i} → A_{i+1}, i = 1, …, 3 subject to the following relations:

(1)  g · f_{1} = f_{1} · r_{1}(g)  g ∈ P (1, 1)
(2)  g · f_{2} = f_{2} · r_{2}(g)  g ∈ P (2, 1)
(3)  g · f_{3} = f_{3} · r_{3}(g)  g ∈ P (3, 1)
(4)  g · f_{2} · f_{1} = f_{2} · f_{1} · r_{1}(g)  g ∈ P (1, 2)
(5)  g · f_{3} · f_{2} = f_{3} · f_{2} · r_{2}(g)  g ∈ P (2, 2)
(6) g · f_{3} · f_{2} · f_{1} = f_{3} · f_{2} · f_{1} · r_{1}(g)  g ∈ P (1, 3)

2.2 Remark: Lemma 2.1 implies that in order to construct a functor F : A → C for some category C, it is enough to find objects F (A_{i}) = C_{i}, i = 1, …, 4 of C, homomorphisms GL(i, F_{2}) → Hom_{C}(C_{i}, C_{i}), and maps F (f_{i}) : C_{i} → C_{i+1}, i = 1, …, 3 such that relations parallel to 2.1(1)–2.1(6) are satisfied. 2.3 A cohomology diagram: Now let V stand for the elementary abelian group (Z/2)^{4}. The natural (left) action of Aut(V) on V induces a right action of Aut(V) on the cohomology ring H^{∗}BV. The ring of elements fixed under this action is the algebra R of rank 4 mod 2 Dickson invariants [32].

If A is an object of A let α̃_{R}(A) = H^{∗}Emb(BA, BV); this gives a functor α̃_{R} from A to the category of right Aut(V)-objects in the category K (1.2). Note (cf. proof of 1.6) that α̃_{R}(A) is isomorphic to the tensor product H^{0} Emb(A, V) ⊗ H^{∗}BV with the diagonal action of Aut(V). Let α_{R} be the fixed point subfunctor of α̃_{R}, so that for each object A_{i} of A, α_{R}(A_{i}) = α̃_{R}(A_{i})^{Aut(V)}.

2.4 Proposition. There is a natural isomorphism R ≅ \underleftarrow{lim} α_{R}. The

higher derived functors \underleftarrow{lim}^{i} α_{R} vanish for i > 0.

Proof: Let Γ = Aut(V) and let M be the right Aut(V) module S = H^{∗}BV treated as a left Γ module via the inversion map of Aut(V). The category A is equivalent to the category A_{Γ} of §8 in such a way that α_{R} corresponds to the functor α_{Γ,M}; proposition 8.1 thus gives the desired result. (Note that in this special case, in which Γ = Aut(V) is as large as possible, the categoryA_{Γ} is equivalent to the category B of 8.3 in such a way that the functor α_{Γ,M} is equivalent to β_{Γ,M}.)

Proposition 2.4 also follows from the main theorem of [13], since in fact the functor α_{R} of 2.4 is naturally equivalent to the functor α_{R} of [13, 1.5].

Suppose that F : A^{op} → T is a functor, where T is the category of topological spaces. According to [3, XII, 5.8] there is a spectral sequence for the cohomology of the homotopy direct limit ho\underrightarrow{lim} F which has the form

\{E_{2}^{p,q} = \underleftarrow{lim}^{p}H^{q}(F)\} ⇒ H^{p+q}(ho\underrightarrow{lim} F) .

If H^{∗}(F) ≅ α_{R}, then by 2.4 the spectral sequence collapses into the isomorphism H^{∗}(ho\underrightarrow{lim} F) ≅ R and by [3, VII, 3.2] the F_{2}-completion of ho\underrightarrow{lim} F is an F_{2}-complete space with cohomology given by R. In this way the following result implies Theorem 1.1.

2.5 Theorem. There exists a functor F : A^{op} → T such that H^{∗}(F) is naturally equivalent to α_{R}.

For the rest of the paper we will concentrate on proving 2.5.

§3. Subgroups of Spin(7)

In §2 we described a very small category A which up to equivalence should be the category of conjugacy classes of elementary abelian subgroups (1.5) of DI(4), and we reduced the problem of constructing BDI(4) to that of constructing a suitable functor F : A^{op} ⟶ T . As explained in 1.5, the spaces which the functor F will take on as its values will be of the form B̂K, where K is the centralizer in Spin(7) of an elementary abelian 2-group containing the center of Spin(7). In this section we take the first steps toward constructing F (or its homotopy version F̄) by identifying (3.2) within Spin(7) the appropriate elementary abelian 2-subgroups E_{i}, (i = 1, …, 4) and collecting needed information about

(1) the centralizers C(E_{i}) of the groups E_{i} (3.3),

(2) the normalizers N (C(E_{i})) of the groups C(E_{i}) (3.5), and

(3) the cohomology rings H^{∗}BC(E_{i}) (3.8). The normalizers (2) determine certain key self-equivalences of the spaces B̂C(E_{i}) (the “internal Spin actions” of §6). In order for us to prove that the functor F̄ constructed in §6 has the correct cohomological behavior, it is necessary to understand the cohomology rings (3) in a functorial, geometric way (3.13).

Let T̄ be the maximal torus in SO(7) given by block matrices

( \begin{matrix}M_{1} & 0 & 0 & 0 \\ 0 & M_{2} & 0 & 0 \\ 0 & 0 & M_{3} & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

where each M_{i} is of the form ( \begin{matrix}cos θ_{i} & − sin θ_{i} \\ sin θ_{i} & cos θ_{i} \end{matrix} ). The Lie algebra of

T̄ is a subalgebra of the Lie algebra so(7) of 7 × 7 skew-symmetric real matrices. Let L̄ denote the kernel of the exponential map on the Lie algebra of T̄; L̄ is a free Z module of rank 3 with basis ℓ̄_{i} (i = 1, …, 3), where

ℓ̄_{i} = ( \begin{matrix}M_{1}^{i} & 0 & 0 & 0 \\ 0 & M_{2}^{i} & 0 & 0 \\ 0 & 0 & M_{3}^{i} & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

with M_{j}^{i} = ( \begin{matrix}0 & −2π \\ 2π & 0 \end{matrix} ) if i = j and M_{j}^{i} = 0 otherwise. Let T be the

torus in Spin(7) which is the inverse image of T̄ and L the kernel of the exponential map on the Lie algebra of T. The natural map L → L̄ exhibits L as the index two subgroup of L̄ consisting of elements n_{1}ℓ̄_{1} + n_{2}ℓ̄_{2} + n_{3}ℓ̄_{3} such that n_{1} + n_{2} + n_{3} is even; this is spanned by the set {ℓ_{1}, ℓ_{2}, ℓ_{3}} where ℓ_{1} = 2ℓ̄_{1}, ℓ_{2} = ℓ̄_{1} − ℓ̄_{2}, and ℓ_{3} = ℓ̄_{1} + ℓ̄_{3}. The group L is naturally isomorphic to the fundamental group π_{1}T.

Recall that the map B Spin(7) → B SO(7) is up to homotopy a principal fibration with fibre K(Z/2, 1) classified by the second Stiefel-Whitney class w_{2} ∈ H^{2}B SO(7). We will implicitly use the following lemma several times.

3.1 Lemma. Let V̄ be an elementary abelian 2-subgroup of SO(7), V its inverse image in Spin(7), and w_{2}(V̄) ∈ H^{2}BV̄ the restriction of w_{2}. Then V is an abelian group iff w_{2}(V̄) is the square of a one-dimensional class and an elementary abelian group iff w_{2}(V̄) = 0.

Remark: Often the best way to compute the class w_{2}(V̄) of 3.1 is to express the representation V̄ → SO(7) as a sum of one-dimensional representations of V̄.

3.2 Elementary abelian subgroups of Spin(7). We will let Ē be the rank three elementary abelian 2-subgroup of SO(7) generated by the matrices diag(−−−−+++), diag(−−++−−+) and diag(−+−+−+−), where diag(⋯) indicates a diagonal matrix with the specified pattern of ±1 entries. By 3.1 the inverse image E_{4} of Ē in Spin(7) is a rank 4 elementary abelian 2-group, and it is clear that E_{4} contains a rank 3 elementary abelian subgroup E_{3} generated by the elements of order 2 in the torus T. The exponential exact sequence L → L⊗R → T shows that E_{3} is naturally isomorphic to (\frac{1}{2}L)/L ≅ L/2L; under this isomorphism the basis {ℓ_{1}, ℓ_{2}, ℓ_{3}} for L provides a basis for E_{3}. Let E_{2} and E_{1} denote the (based) subgroups of E_{3} corresponding to the subgroups of L/2L generated respectively by {ℓ_{1}, ℓ_{2}} and {ℓ_{1}}. The group E_{1} is the center of Spin(7), i.e. the kernel of the epimorphism Spin(7) → SO(7).

3.3 Centralizers of elementary abelian subgroups. We need to describe the centralizers of the above groups very explicitly (see 1.5). Let Q stand for the multiplicative group of unit quaternions; this is isomorphic to the special unitary group SU(2). The center of Q is {±1}, and we will use Q_{red}^{n} to stand for the quotient of Q^{n} by the diagonal subgroup generated by (−1, …, −1). Let T_{ext}^{n} denote the semidirect product of the n-torus (R/Z)^{n} with {±1}, where the generator of {±1} acts on (R/Z)^{n} by inversion. Recall (1.9) that C(E_{i}) (resp. N (E_{i})) is the centralizer (resp. normalizer) in Spin(7) of E_{i}.

3.4 Lemma. The groups C(E_{i}), i = 1, …, 4 are isomorphic respectively to Spin(7), Q_{red}^{3}, T_{ext}^{3}, and E_{4}. The quotient group N (E_{4})/C(E_{4}) has order 8 · |GL(3, F_{2})|.

Proof: We will use the fact that if K ⊂ Spin(7) is the inverse image of the subgroup K̄ ⊂ SO(7), then the normalizer of K in Spin(7) is the inverse image of the normalizer of K̄. Case i = 1 of the lemma is clear. For i = 2, note that the image of E_{2} in SO(7) is the group Ē_{2} of order 2 generated by diag(−−−−+++). The normalizer N (Ē_{2}) of Ē_{2} in SO(7) is a two-component group with identity component SO(4)×SO(3); one element in the non-identity component is diag(−−−+++−). By 3.1, C(E_{2}) is the inverse image in Spin(7) of the identity component of N (Ē_{2}), and it is easy to check that this inverse image, as a two-fold covering group of SO(4) × SO(3) ≅ Q_{red}^{2} × Q_{red}^{1}, has the indicated form.

For i = 3 observe that the image Ē_{3} of E_{3} in SO(7) is the group generated by Ē_{2} and diag(−−++−−+); the representation Ē_{3} → SO(7) is the sum of the trivial one-dimensional representation of Ē_{3} with two copies of each non-trivial one-dimensional representation. The centralizer C(Ē_{3}) of Ē_{3} in SO(7) is a group isomorphic to O(2)^{3} generated by the torus T̄ and the rank 6 elementary abelian group D of diagonal matrices. A check with 3.1 shows that C(E_{3}) is generated by the torus T and the inverse image in Spin(7) of diag(−+−+−+−).

For i = 4, observe that the image Ē_{4} of E_{4} in SO(7) is generated by Ē_{3} and diag(−+−+−+−); the representation Ē_{4} → SO(7) is the sum of all seven non-trivial one-dimensional representations of Ē_{4}. The centralizer C(Ē_{4}) is generated by Ē_{4} and the group D above; the quotient N (Ē_{4})/C(Ē_{4}) is isomorphic in its conjugation action on Ē_{4} to Aut(Ē_{4}). It is now easy to see that E_{4} is its own centralizer in Spin(7), since it is clear from the above that E_{4} is its own centralizer in C(E_{3}). The desired result follows.

3.5 Normalizers of centralizers. We now study the normalizers in Spin(7) of each one of above centralizers. This is important to us because (3.6) the normalizers determine which self homotopy equivalences of the classifying spaces of the centralizers are realized by conjugations in Spin(7). In the end (see 1.7) we will have to augment these particular self homotopy equivalences by others in order to construct the functor F̄, but it is necessary (§6) in verifying the commutative diagrams that constitute F̄ to recognize that certain equivalences are internal to Spin(7). We approach the problem of naming the equivalences (3.7) by labeling them with integral matrices (see 5.5 and the proof of 6.1 for an indication of how effective this is).

Note that for i = 1, …, 4 the group E_{i} is the center Z(C(E_{i})), so that the normalizer N (E_{i}) of E_{i} in Spin(7) is also the normalizer N (C(E_{i})) of C(E_{i}) in Spin(7). For i = 1, 2, 3 let W_{i} be the Weyl group of C(E_{i}) (i.e., the quotient by T of the normalizer of T in C(E_{i})), and let W_{i}^{+} be the quotient by T of the normalizer of T in N (E_{i}). The groups W_{i} and W_{i}^{+} are all in a natural way subgroups of the Weyl group W_{1} of Spin(7); in fact there are inclusions W_{3} ⊂ W_{2} ⊂ W_{2}^{+} ⊂ W_{3}^{+} = W_{1} = W_{1}^{+}. 3.6 Remark: Since inner automorphisms of a group G act trivially up to homotopy on BG, there are natural homotopy actions of W_{i}^{+}/W_{i} on BC(E_{i}), i = 2, 3.

Given the chosen basis {ℓ_{1}, ℓ_{2}, ℓ_{3}} for L, the conjugation action of W_{1} on T provides a homomorphism ρ_{W} : W_{1} → GL(3, Z). Let μ : GL(3, Z) → GL(3, F_{2}) stand for reduction mod 2 and det : GL(3, Z) → {±1} for the determinant map. Recall from §2 the definition of the subgroup P (1, 2) of GL(3, F_{2}). Denote by U(2, 1) the subgroup of P (2, 1) consisting of matrices which agree with the identity except in the last column and by U ⊂ GL(3, F_{2}) the group of upper-triangular matrices.

3.7 Lemma. The product map ϕ = (μ · ρ_{W} , det ·ρ_{W}) induces isomorphism W_{1} ≅ P (1, 2) × {±1}, W_{2}^{+} ≅ U × {±1}, W_{2} ≅ U(2, 1) × {±1}, and W_{3} ≅ {±1}. Under these isomorphisms the map ρ_{W} sends −1 ∈ {±1} to the negative of the identity matrix in GL(3, Z).

Proof: The group Σ_{3} ≀ {±1} acts on L̄ by the formula

σ · [ϵ_{1}, ϵ_{2}, ϵ_{3}] · (a_{1}ℓ̄_{1} + a_{2}ℓ̄_{2} + a_{3}ℓ̄_{3}) = ϵ_{1}a_{1}ℓ̄_{σ(1)} + ϵ_{2}a_{2}ℓ̄_{σ(2)} + ϵ_{3}a_{3}ℓ̄_{σ(3)}

= ϵ_{σ^{−1}(1)}a_{σ^{−1}(1)}ℓ̄_{1} + ϵ_{σ^{−1}(2)}a_{σ^{−1}(2)}ℓ̄_{2} + ϵ_{σ^{−1}(3)}a_{σ^{−1}(3)}ℓ̄_{3}

where σ ∈ Σ_{3}, ϵ_{i} = ±1 and a_{i} ∈ Z. Under the isomorphism π_{1}T̄ ≅ L̄ this is the conjugation action of the Weyl group of SO(7) on L̄. The action restricts to an action of W_{1} ≅ Σ_{3} ≀ {±1} on the sublattice L of L̄, and the stated formula for ϕ(W_{1}) can be derived by a straightforward calculation. The rest of the lemma follows from the way in which the groups involved are defined. For example, W_{1} acts on E_{3} by conjugation and it is clear that W_{2} is the subgroup of W_{1} consisting of elements which pointwise fix the first two basis elements of E_{3}. Since W_{1} acts on E_{3} ≅ L/2L as P (1, 2), the desired identification W_{2} follows immediately.

3.8 Cohomology of centralizers. To conclude this section we will give a description (3.13) of the cohomology rings of the spaces BC(E_{i}) which is functorial enough for the purposes of §6. All of these cohomology rings are rings of invariants (3.11).

3.9 Lemma. (Lannes) Let G be a compact Lie group, E ⊂ G an elementary abelian 2-subgroup and W′ the quotient N (E)/C(E). Suppose that the natural restriction map H^{∗}BG → (H^{∗}BE)^{W′} is an isomorphism. Let V be an elementary abelian 2-group. Then in the natural commutative square

Hom(V, E)/W′  ⟶ Hom(V, G)/G
↓  ↓
(π_{0}Map(BV, BE))/W′ ⟶ π_{0}Map(BV, BG)

all of the arrows are bijections. Moreover if h : BV → BG is a homotopy class of maps, k : BV → BE is a homotopy class covering h, and W′(k) ⊂ W′ is the subgroup of elements w such that w · k = k, then the evident restriction map H^{∗}Map(BV, BG)_{h} → (H^{∗}Map(BV, BE)_{k})^{W′(k)} is an isomorphism.

Remark: Here Hom(V, E)/W′ denotes the collection of orbits of the action of W′ on the set Hom(V, E), and Hom(V, G)/G is the set of conjugacy classes of homomorphisms V → G. Note that for any k : BV → BE evaluation at the basepoint gives an equivalence Map(BV, BE)_{k} → BE which is equivariant with respect to W′(k); this equivalence induces the isomorphism (H^{∗}Map(BV, BE)_{k})^{W′(k)} ≅ (H^{∗}BE)^{W′(k)}. Proof of 3.9: This follows from 1.6 and the fact [20] [14] that for any compact Lie group G and elementary abelian 2-group V the map

T^{V} H^{∗}(BG) ⟶ H^{∗}Map(BV, BE)

is an isomorphism.

For i = 1, …, 4 let W′_{i} denote the quotient group N_{i}(E_{4})/E_{4}, where N_{i}(E_{4}) is the normalizer of E_{4} in C(E_{i}). Choose a basis for E_{4} which extends the basis given above for E_{3}, so that the conjugation action of W′_{i} on E_{4} gives a homomorphism W′_{i} → GL(4, F_{2}). Let P_{0}(i, j) denote the kernel of the map r_{i} : P (i, j) → GL(i, F_{2}).

3.10 Lemma. The conjugation action of W′_{i} on E_{4} (i = 1, …, 4) induces an isomorphism W′_{i} ≅ P_{0}(i, 4 − i).

Proof: The map W′_{i} → GL(4, F_{2}) is clearly a monomorphism; it has image contained in P_{0}(i, 4 − i) because the first i basis vectors of E_{4} belong to the center of C(E_{i}). Case 1 is proved by observing (3.4) that W′_{1} and P_{0}(1, 3) have the same order. The rest follows from the fact that essentially by definition W′_{i} (2 ≤ i ≤ 4) is the group of elements in W′_{1} which fix the first i basis vectors of E_{4}.

3.11 Lemma. For i = 1, …, 4 the restriction map H^{∗}BC(E_{i}) → H^{∗}BE_{4} induces an isomorphism H^{∗}BC(E_{i}) ≅ (H^{∗}BE_{4})^{W′_{i}}.

Proof: We will first treat the case i = 1. According to [2] or [28] the algebra H^{∗}B Spin(7) is a polynomial algebra on classes x_{4}, x_{6}, x_{7}, x_{8}. The restriction map H^{∗}B Spin(7) → H^{∗}BE_{4} makes H^{∗}BE_{4} a finitely generated module over H^{∗}B Spin(7) [29, 2.4] and, since H^{∗}BE_{4} is itself a polynomial algebra on four (one-dimensional) generators, it follows from a transcendence degree argument [1, 11.21] that this restriction map must be injective. The ring H^{∗}BE_{4} is integral [1, 5.1] over H^{∗}B Spin(7) (this is an easy consequence of the fact that H^{∗}B Spin(7) is Noetherian and H^{∗}BE_{4} is finitely generated as a module over H^{∗}B Spin(7)). Since the product of the degrees of the polynomial generators of H^{∗}B Spin(7) equals the order of W′_{1}, the desired isomorphism follows from [32, 3.2].

Cases i = 2, 3, 4 now follow from 3.9, since BC(E_{i}) is naturally equivalent [14] at the prime 2 to Map(BE_{i}, B Spin(7))_{f}, where the map f : BE_{i} → B Spin(7) is induced by the inclusion E_{i} → Spin(7).

Let δ_{i} : BE_{i} → BC(E_{i}) be induced by the inclusion E_{i} → C(E_{i}).

3.12 Lemma. Let V be a rank 4 elementary abelian 2-group and let h : E_{i} → V be a monomorphism (i ∈ {1, …, 4}). Then up to homotopy there is a unique map k ∈ Emb(BV, BC(E_{i})) such that k · Bh is homotopic to δ_{i}.

Proof: This is an immediate consequence of 3.11, 3.9 and 3.10, since up to the action of W′_{i} there is only one monomorphism V → E_{4} which when composed with h gives the inclusion E_{i} → E_{4}. Note that by 1.6(2) the bijection (3.9) Hom(V, E_{4})/W′_{i} ≅ π_{0}Map(BV, BC(E_{i})) restricts to a bijection Emb(V, E_{4})/W′_{i} ≅ π_{0}Emb(BV, BC(E_{i})).

Now let V be a rank 4 elementary abelian 2-group and choose i ∈ {1, …, 4}. Lemma 3.12 shows that up to homotopy there is a unique map

Emb(BE_{i}, BV) ≅ Emb(E_{i}, V) × BV ⟶^{κ_{i,V}} BC(E_{i})

with the property that for each f ∈ Emb(BE_{i}, BV) the diagram

BE_{i} ≅ Aut(E_{i})_{1} ⟶^{δ_{i}} BC(E_{i})
f·(–)↓  ↓^{=}
Emb(BE_{i}, BV) ⟶^{κ_{i,V}} BC(E_{i})

homotopy commutes and such that κ_{i,V}^{∗} makes H^{∗}Emb(BE_{i}, BV) a finitely generated module over H^{∗}BC(E_{i}). (The equivalence Aut(E_{i})_{1} ≅ BE_{i} in the above diagram is obtained by evaluation at the basepoint BE_{i}, as is each equivalence Emb(BE_{i}, BV) ≅ BV.) By uniqueness the map κ_{i,V} is equivariant with respect to the evident action of Aut(V) on its domain and the trivial action of Aut(V) on its range. It follows that the cohomology map κ_{i,V}^{∗} induces a map from H^{∗}BC(E_{i}) to the fixed point set of the action of Aut(V) on H^{∗}Emb(BE_{i}, BV).

3.13 Proposition. Let V be a rank 4 elementary abelian 2-group. Then for each i ∈ {1, …, 4} the map

H^{∗}BC(E_{i}) → (H^{∗}Emb(BE_{i}, BV))^{Aut(V)}

induced by κ_{i,V}^{∗} is an isomorphism.

Proof: Let h : E_{i} → V be a monomorphism and k : BV → BC(E_{i}) the extension of 3.12. Let W′ ⊂ Aut(V) be the subgroup consisting of elements w such that k · Bw is homotopic to k. By Shapiro’s lemma the desired statement amounts to a claim that the map H^{∗}BC(E_{i}) → (H^{∗}BV)^{W′} is an isomorphism. This follows from 3.10 and 3.11.

§4. A 2-adic representation

In this section we will show that the conjugation action of the Weyl group of Spin(7) on the fundamental group of the torus actually extends (after tensoring with Ẑ_{2}) to an action of a larger group. This will allow us in §6 to construct an action of GL(3, F_{2}) on the space B̂C(E_{3}) (see 3.4).

Under the isomorphism W_{1} ≅ P (1, 2) × {±1} provided by 3.7, the homomorphism ρ_{W} gives a map ρ_{P} : P (1, 2) → GL(3, Z). Let ρ̂_{P} denote the composite of ρ_{P} with the natural inclusion GL(3, Z) → GL(3, Ẑ_{2}) and μ̂ : GL(3, Ẑ_{2}) → GL(3, F_{2}) the map given by reduction mod 2.

4.1 Theorem. There is a homomorphism ρ̂_{GL} : GL(3, F_{2}) → GL(3, Ẑ_{2}) such that

(1) the composite μ̂ · ρ̂_{GL} is the identity map of GL(3, F_{2}), and

(2) the restriction of ρ̂_{GL} to P (1, 2) is ρ̂_{P}.

4.2 Remark: By construction the composition μ̂ · ρ̂_{P} gives the identity map of P (1, 2). Theorem 4.1 leads immediately to a homomorphism GL(3, F_{2}) × {±1} → GL(3, Ẑ_{2}) which in an evident sense (3.7) extends ρ_{W}.

Let sl(3, F_{2}) denote the additive group of trace zero 3 × 3 matrices over F_{2}. The group GL(3, F_{2}) acts on sl(3, F_{2}) by conjugation.

4.3 Lemma. For any subgroup K of GL(3, F_{2}) and i ≥ 1 the cohomology group H^{i}(K, sl(3, F_{2})) vanishes.

Proof: Let U ⊂ GL(3, F_{2}) be the group of upper-triangular matrices; the group U has order 8 and is a 2-Sylow subgroup of GL(3, F_{2}). A calculation shows that the vector space H_{0}(U, sl(3, F_{2})) is of dimension one. Let x ∈ sl(3, F_{2}) be an element which projects to a generator of H_{0}(U, sl(3, F_{2})) and let f : F_{2}[U] → sl(3, F_{2}) be the F_{2}[U] module map which sends 1 to x. Since the augmentation ideal of F_{2}[U] is nilpotent [4, VI, 8.3] Nakayama’s lemma [4, VI, 8.4] shows that the map f is surjective. By a dimension count, then, f is an isomorphism. Now let K ⊂ GL(3, F_{2}) be any subgroup and K_{2} ⊂ K its 2-Sylow subgroup; note that K_{2} is conjugate in GL(3, F_{2}) to a subgroup of U. It is clear that sl(3, F_{2}) is free as a module over F_{2}[K_{2}] and hence [4, VI, 8.5] that the group H^{i}(K_{2}, sl(3, F_{2})) vanishes for i ≥ 1. The lemma follows from a transfer argument [4, III, 10.3].

4.4 Lemma. Suppose that ι : K → GL(3, F_{2}) = SL(3, F_{2}) is a subgroup inclusion. Then

(1) there exists a group homomorphism ρ : K → SL(3, Ẑ_{2}) such that μ̂ · ρ = ι, and

(2) if ρ′ is another group homomorphism K → SL(3, Ẑ_{2}) with μ̂ · ρ = ι, then there exists an element C ∈ SL(3, Ẑ_{2}) such that μ̂(C) is the identity matrix and CρC^{−1} = ρ′.

Proof: Let G denote SL(3, Ẑ_{2}) and let G_{n} ⊂ G (n ≥ 1) be the normal subgroup consisting of matrices which are congruent to the identity matrix modulo 2^{n}. It is easy to see that for each n ≥ 1 the quotient group G_{n}/G_{n+1} is abelian and is isomorphic as a module over G/G_{n} to sl(3, F_{2}), where the action of G/G_{n} on sl(3, F_{2}) is obtained by composing the quotient map G/G_{n} → GL(3, F_{2}) with the conjugation action that appears in 4.3.

Suppose that ι has been lifted to a homomorphism ρ_{n} : K → G/G_{n} for some n, and let q : K′ → K be the pullback over ρ_{n} of the surjection G/G_{n+1} → G/G_{n}. The kernel of q is an abelian group which is isomorphic as a K module to sl(3, F_{2}). By 4.3 the group H^{2}(K, sl(3, F_{2})) vanishes and so [4, IV, 3.12] the epimorphism q can be split; moreover, since H^{1}(K, sl(3, F_{2})) also vanishes, any two such splittings are conjugate by an element in the kernel of q [4, IV, 2.3]. It follows that the map ρ_{n} lifts to a homomorphism ρ_{n+1} : K → G/G_{n+1} and that, if ρ′_{n+1} is another such lift, the maps ρ_{n+1} and ρ′_{n+1} are conjugate by an element in the kernel of G/G_{n+1} → G/G_{n}.

By induction, then, the map ι lifts for any n to a homomorphism ρ_{n} : K → G/G_{n}, and any two such lifts are conjugate via an element C_{n} ∈ G/G_{n} which projects to the identity element of GL(3, F_{2}). Let L(n) be the set {ρ_{n}} of all such lifts. There are natural surjections L(n) → L(n − 1), and L(∞) = \underleftarrow{lim} L(n) is the set of all homomorphisms ρ : K → G such that μ̂ · ρ = ι. It is clear that L(∞) is nonempty. This proves (1).

Now suppose that ρ and ρ′ are elements of L(∞). Let ρ_{n} and ρ′_{n} denote the images of these homomorphisms in L(n), and let C(n) denote the set of all elements of G/G_{n} which conjugate ρ_{n} to ρ′_{n} and project trivially to GL(3, F_{2}). There are natural maps C(n) → C(n − 1), and C(∞) = \underleftarrow{lim} C(n) is the set of all elements of G which conjugate ρ to ρ′ and project trivially to GL(3, F_{2}). Part (2) of the lemma follows from the fact that C(∞) is nonempty, since each C(n) is finite and nonempty. Proof of 4.1: By part (1) of 4.4 applied to K = GL(3, F_{2}), there exists a homomorphism ρ̂_{GL} : GL(3, F_{2}) → GL(3, Ẑ_{2}) which satisfies (1) of 4.1. By part (2) of 4.4 applied to K = P (1, 2) (see 4.2), the restriction of ρ̂_{GL} to P (1, 2) is conjugate to ρ̂_{P} via an element C ∈ GL(3, Ẑ_{2}) which projects trivially to GL(3, F_{2}). The proof is completed by replacing ρ̂_{GL} with its conjugate by C.

§5. Self-maps of completed classifying spaces

In this section we will calculate (5.5) the space of self homotopy equivalences of B̂Q_{red}^{n} (see 3.4). This will allow us in §6 to construct an action of GL(2, F_{2}) on the space B̂C(E_{2}) ≅ B̂Q_{red}^{3}. There is an evident permutation action of GL(2, F_{2}) ≅ Σ_{3} on B̂Q_{red}^{3}, but the action which comes up in §6 is another one.

Some of what we do is implicit in [14], [10] [27], although the particular results we need don’t appear in these references. For the rest of the section let G denote a compact Lie group with chosen maximal torus T and Weyl group W.

The set Emb(T, G) (see 1.9) has a distinguished basepoint given by the inclusion of T in G. Since T is unique up to conjugacy the discrete group Aut(T) acts transitively by composition on Emb(T, G); it is clear that the isotropy group of the distinguished basepoint is the image of the map W → Aut(T) given by the conjugation action of W on T. We will need a homotopy theoretic version of this fact. Note that the set π_{0}Emb(B̂T, B̂G) has a distinguished basepoint given by the map B̂T → B̂G induced by the inclusion T → G, and that the discrete group π_{0}Aut(B̂T) acts on π_{0}Emb(B̂T, B̂G) by composition.

5.1 Proposition. If G is a connected compact Lie group then the natural action of π_{0}Aut(B̂T) on π_{0}Emb(B̂T, B̂G) is transitive; the isotropy group of the distinguished basepoint is the image of the map W → π_{0}Aut(B̂T) given by the conjugation action of W on T.

5.2 Remark: This will be proven below. Note that choice of a basis for π_{1}T gives isomorphisms Aut(T) ≅ GL(r, Z) and π_{0}Aut(B̂T) ≅ GL(r, Ẑ_{2}), where r is the rank of G.

5.3 Lemma. Suppose that G is connected and that π is a finite twogroup. Then

(1) the natural map Map(Bπ, BG) → Map(Bπ, B̂G) induces an isomorphism on mod 2 cohomology, and

(2) the space Map(Bπ, B̂G) is F_{2}-complete in the sense of [3].

Remark: Lemma 5.3 implies that Map(Bπ, B̂G) is homotopy equivalent to the F_{2}-completion of Map(Bπ, BG) .

Proof of 5.3: Part (1) is proved by the argument of [6, 4.5]; the main step is to note that the homotopy fibre of the map BG → B̂G is a simple space with uniquely 2-divisible homotopy groups. Part (2) now follows from [3, Ch. VII, 5.1] and the fact [14, 1.1] that each component of Map(Bπ, BG) has a finite fundamental group.

Proof of 5.1: Let T_{n} be the subgroup of T consisting of elements x such that x^{2^{n}} = 1, and let T_{∞} denote the discrete group ∪_{n}T_{n}. The composite map BT_{∞} → BT → B̂T is a cohomology isomorphism and so induces an isomorphism π_{0}Emb(B̂T, B̂G) ≅ π_{0}Emb(BT_{∞}, B̂G). It is easy to see by explicit calculation that the functorial map Aut(T_{∞}) → π_{0}Aut(B̂T_{∞}) ≅ π_{0}Aut(B̂T) is an isomorphism. To prove 5.1, then, it is enough to show that Aut(T_{∞}) acts transitively on π_{0}Emb(BT_{∞}, B̂G) and that the isotropy group of the evident distinguished basepoint is the image of the map W → Aut(T_{∞}) given by the conjugation action of W on T_{∞}.

Since BT_{∞} is equivalent to the homotopy direct limit ho\underrightarrow{lim} BT_{n} [3,

XII, 3.6], the mapping space Emb(BT_{∞}, B̂G) is equivalent to the homotopy inverse limit ho\underleftarrow{lim} Emb(BT_{n}, B̂G) (see [3, XII, 4.1]). Choose n_{0} >> 0 so that for n > n_{0}

(1) the map Emb(T_{n}, T) → Emb(T_{n}, G) is surjective, and

(2) the centralizer of T_{n} in G is exactly T. (It is easy to obtain (1) for large n by observing that any f : T_{n} → G has image contained up to conjugacy in the normalizer N (T) of T [30] and then using the fact that W = T/N (T) is a finite group. For (2), let C_{n} be the centralizer in G of T_{n}, observe that T is self-centralizing and T_{∞} is dense in T so that ∩_{n}C_{n} = T, and use a dimension argument to conclude that C_{n} = T for sufficiently large n). By [14, 1.1] and 5.3 the natural map Emb(T_{n}, G) → π_{0}Emb(BT_{n}, B̂G) is an isomorphism for any n and for n > n_{0} each component of Emb(BT_{n}, B̂G) is naturally homotopy equivalent to B̂T. It follows [3, IX, 3.5] that \underleftarrow{lim}^{1} π_{1}Emb(BT_{n}, B̂G) vanishes for every choice of a compatible sequence of basepoints and [3, XI, 4.1; IX 3.1] that there is an isomorphism π_{0}Emb(BT_{∞}, B̂G) ≅ \underleftarrow{lim} Emb(T_{n}, G). The desired result can now be proven by combining the isomorphism Aut(T_{∞}) ≅ \underleftarrow{lim} Aut(T_{n}) with the fact that for n > n_{0} the action of Aut(T_{n}) on Emb(T_{n}, G) is clearly transitive and has the conjugation image of W in Aut(T_{n}) as the isotropy group of the natural basepoint.

Now suppose that G is connected. Since T is self-centralizing the conjugation action of W on B̂T is faithful, and we will use this action to identify W with a subgroup of π_{0}Aut(B̂T). Let ν = ν_{G} denote the normalizer of W in π_{0}Aut(B̂T) and ν̄ = ν̄_{G} the quotient of ν by W. By 5.1, ν̄ is isomorphic to the group of automorphisms of π_{0}Emb(B̂T, B̂G) as a set with an action of π_{0}Aut(B̂T). Under this isomorphism, the left composition action of π_{0}Aut(B̂G) on π_{0}Emb(B̂T, B̂G) gives a map χ = χ_{G} : π_{0}Aut(B̂G) → ν̄.

Recall that Z(G) denotes the center of the compact Lie group G. As in [14, §1] the group homomorphism Z(G) × G → G gives a map BZ(G) → Aut(B̂G)_{1}.

5.4 Lemma. The space Map(B̂Q, B̂Q) is the union of Aut(B̂Q) with the component Map(B̂Q, B̂Q)_{0} of null homotopic maps. Moreover

(1) the map χ_{Q} : π_{0}Aut(B̂Q) → ν̄_{Q} is an isomorphism,

(2) the natural map BZ(Q) → Aut(B̂Q)_{1} is an equivalence, and (3) the basepoint evaluation map Map(B̂Q, B̂Q)_{0} → B̂Q is an equiv-

alence.

Remark: The group Z(Q) is {±1} and the group ν̄_{Q} is canonically isomorphic to Ẑ_{2}^{×}/{±1}.

Proof: Let Õ_{48} and Q_{16} be the subgroups of Q constructed in [8, §4] and let N_{2}T ⊂ Q be the subgroup of the normalizer of the torus in Q consisting of elements with order a power of 2. It follows from [8, 4.1] that there is a diagram BÕ_{48} ← BQ_{16} → BN_{2}T induced by subgroup inclusions and a map from the homotopy pushout P of this diagram to B̂Q which induces an isomorphism on cohomology. As a consequence the space Map(B̂Q, B̂Q) ≅ Map(P, B̂Q) is equivalent to the homotopy pullback of the induced diagram

Map(BÕ_{48}, B̂Q) → Map(BQ_{16}, B̂Q) ← Map(BN_{2}T, B̂Q) .

By [14, 1.1], 5.3, and a limiting argument based on the fact that N_{2}T is a union of finite two-groups (cf. proof of 5.1), it is possible to identify Map(BN_{2}T, B̂Q) up to homotopy as a disjoint union involving one copy of B̂Q corresponding to the component of null homotopic maps together with multiple copies of B{±1} (one for each element ν̄_{Q}). Similarly, the subspace of Map(BQ_{16}, B̂Q) consisting of maps which extend up to homotopy to BN_{2}T is up to homotopy the disjoint union of B̂Q with two copies of B{±1}. Finally, both Q_{16} and Õ_{48} contain a normal subgroup Q_{8} of order 8 (essentially the classical quaternion group); the subspace M_{e} of M = Map(BQ_{8}, B̂Q) consisting of maps which extend up to homotopy to BN_{2}T is up to homotopy the disjoint union of B̂Q with one copy of B{±1}. As in [14, 5.1] there are equivalences

Map(BÕ_{48}, B̂Q) ≅ M^{hγ}
Map(BQ_{16}, B̂Q) ≅ M^{hγ′}

where we have used the homotopy fixed point set notation (–)^{h-} of [14], γ = Õ_{48}/Q_{8} and γ′ = Q_{16}/Q_{8}. Now γ is isomorphic to the symmetric group Σ_{3} and so has a normal subgroup γ′′ of order 3 with γ/γ′′ ≅ γ′. It is easy to see that γ′′ acts trivially up to homotopy on M_{e} and thus by the argument of [14, 2.3] that M_{e}^{hγ′′} ≅ M_{e}. By the argument of [14, 5.1] there is an equivalence M_{e}^{hγ} ≅ (M_{e}^{hγ′′})^{hγ′}. It follows that M_{e}^{hγ} ≅ M_{e}^{hγ′}, which implies that the above homotopy pullback collapses to an equivalence Map(B̂Q, B̂Q) ≅ Map(BN_{2}T, B̂Q). The desired result follows directly.

5.5 Proposition. Suppose that G is Q^{n} or Q_{red}^{n} (n > 0). Then

(1) the map χ_{G} : π_{0}Aut(B̂G) → ν̄_{G} is an isomorphism, and

(2) the natural map BZ(G) → Aut(B̂G)_{1} is an equivalence.

Proof: It is clear that evaluation at the basepoint gives an equivalence Map(B̂Q, BZ(Q)) → BZ(Q). It follows from an inductive argument using 5.4 and the formula

Map(B̂Q^{n}, B̂Q) ≅ Map(B̂Q, Map(B̂Q^{n−1}, B̂Q))

that any map B̂Q^{n} → B̂Q factors up to homotopy through one of the n evident projections and that via these projections Map(B̂Q^{n}, B̂Q) is equivalent to an amalgamated union of n copies of Map(B̂Q, B̂Q) (the amalgamation amounts to identifying the subspaces of null homotopic maps). An argument along the lines of [16, §4] now shows that the obvious map Σ_{n} ≀ Aut(B̂Q) → Aut(B̂Q^{n}) is an equivalence. Since ν̄_{Q^{n}} ≅ Σ_{n} ≀ ν̄_{Q} and Z(Q^{n}) ≅ Z(Q)^{n} the desired result follows easily in the case G = Q^{n}.

For the case G = Q_{red}^{n}, let K denote the Eilenberg-MacLane space K(Z/2, 2) and note that there is a fibration p : B̂Q_{red}^{n} → K (with fibre B̂Q^{n}) which is the projection of B̂Q_{red}^{n} to its second Postnikov stage. Since the formation of Postnikov stages is functorial, the topological monoid of self-equivalences of p in the sense of [7, 6.1] is equivalent to Aut(B̂Q_{red}^{n}) (cf. [7, 6.3]). By [7, 6.2], then, there is a natural weak equivalence

BAut(B̂Q_{red}^{n}) ≅ EAut(K) ×_{Aut(K)} Map(K, BAut(B̂Q^{n}))_{p}

(where the subscript p on the right denotes the component of the indicated mapping space which contains a classifying map for p). Since Aut(K) ≅ K and Map(K, BAut(B̂Q^{n}))_{p} ≅ BAut(B̂Q^{n}) (both via basepoint evaluation maps), it is not hard to calculate directly that the natural map Z(Q_{red}^{n}) → π_{2}BAut(B̂Q_{red}^{n}) is an isomorphism. It is also clear that every self-equivalence of B̂Q_{red}^{n} is covered up to homotopy by a unique self-equivalence of B̂Q^{n}. The proof is completed by checking that if T̃ is a maximal torus of Q^{n} which covers the maximal torus T of Q_{red}^{n}, then each automorphism of T which normalizes the action of the Weyl group of Q_{red}^{n} lifts to a unique automorphism of T̃ which normalizes the action of the Weyl group of Q^{n}.

§6. A construction in the homotopy category

Let Ho(T ) be the homotopy category of T . In this section we will define a functor F̄: A^{op} → Ho(T ) such that H^{∗}F̄ is naturally equivalent to the functor α_{R} of 2.5. We will continue to use the notation of §3.

By construction each vector space E_{i}, i = 1, …, 4 comes with a basis and therefore an action of GL(i, F_{2}); we will refer to the induced action of GL(i, F_{2}) on BE_{i} as the basis action of GL(i, F_{2}) on BE_{i}. By 3.6 and 3.7 the internal structure of Spin(7) provides actions up to homotopy of P (1, 1) ≅ W_{2}^{+}/W_{2} on B̂C(E_{2}) and of P (1, 2) ≅ W_{3}^{+}/W_{3} on B̂C(E_{3}); we will refer to these as the internal Spin actions of these groups. Let δ̂_{i} : BE_{i} → B̂C(E_{i}) be the map induced by the inclusion E_{i} → C(E_{i}).

6.1 Proposition. There are homotopy actions of GL(i, F_{2}) on B̂C(E_{i}), i = 2, 3 with the following properties.

(1) The map δ̂_{i} : BE_{i} → B̂C(E_{i}) is equivariant with respect to the basis action of GL(i, F_{2}) on BE_{i} for i = 2, 3.

(2) The restriction to P (1, 2) of the action of GL(3, F_{2}) agrees with the internal Spin action of P (1, 2) on B̂C(E_{3}).

(3) The restriction to P (1, 1) of the action of GL(2, F_{2}) agrees with

the internal Spin action of P (1, 1) on B̂C(E_{2}). Restricting the action of GL(3, F_{2}) to P (2, 1) gives an action of P (2, 1) on B̂C(E_{3}) and composing the action of GL(2, F_{2}) with the natural projection P (2, 1) → P (2, 1)/U(2, 1) ≅ GL(2, F_{2}) gives an action of P (2, 1) on B̂C(E_{2}).

(4) The map B̂C(E_{3}) → B̂C(E_{2}) induced by the inclusion C(E_{3}) → C(E_{2}) is equivariant with respect to these actions of P (2, 1).

Proof: Let χ_{i} : π_{0}Aut(B̂C(E_{i})) → ν̄_{i} be the map χ_{G} of §5 for G = C(E_{i}), i = 2, 3. The map χ_{2} is an isomorphism by 5.5. It is easy to see that χ_{3} is well-defined (in spite of the fact that C(E_{3}) is not connected) and induces an isomorphism from π_{0}Aut(B̂C(E_{3})) to the quotient of GL(3, Ẑ_{2}) by the subgroup {±I_{3}} generated by the negative of the identity matrix. (Here and in what follows we use the specific basis {ℓ_{1}, ℓ_{2}, ℓ_{3}} of L ≅ π_{2}BT to identify π_{0}Aut(B̂T) with GL(3, Ẑ_{2}).) The representation ρ̂_{GL} of 4.1 can be composed with the quotient map GL(3, Ẑ_{2}) → GL(3, Ẑ_{2})/{±I_{3}} and the inverse of χ_{3} to give an action of GL(3, F_{2}) on B̂C(E_{3}). Condition (1) (i = 3) follows from the isomorphism E_{3} ≅ L/2L together with 4.1(1), condition (2) from 3.7 and 4.1(2).

The group U(2, 1) is a subgroup of P (1, 2). By 3.7 the image of W_{2} in GL(3, Ẑ_{2}) is generated by {±I_{3}} and (4.1) ρ̂_{P} (U(2, 1)), where by 4.1(2) this last group is equal to ρ̂_{GL}(U(2, 1)), Since U(2, 1) is a normal subgroup of P (2, 1), it follows that ρ̂_{GL}(P (2, 1)) is contained in the normalizer of the image of W_{2} and hence that ρ̂_{GL} induces a map P (2, 1)/U(2, 1) ≅ GL(2, F_{2}) → ν̄_{2} ≅ π_{0}Aut(B̂C(E_{2})). Use this map to give an action of GL(2, F_{2}) on B̂C(E_{2}). Condition (1) (i = 2) follows from 4.1(1) and the natural isomorphism between E_{2} and the subgroup of L/2L generated by the residue classes of ℓ_{1} and ℓ_{2}.

Restricting the above action of GL(2, F_{2}) on B̂C(E_{2}) to P (1, 1) and applying χ_{2} gives an monomorphism

P (1, 1) = U/U(2, 1) → ν̄_{2} = N (ρ̂_{GL}(U(2, 1)))/ρ̂_{GL}(U(2, 1))

(see 3.7) which is covered by the map U → GL(3, Ẑ_{2}) given by the restriction to U of ρ̂_{GL}. The internal Spin action of P (1, 1) on B̂C(E_{2}) induces a parallel monomorphism P (1, 1) → ν̄_{2} which is covered by the map U → GL(3, Ẑ_{2}) given by the restriction to U of ρ̂_{P} (3.7, 4.1). Condition (3) is thus a consequence of 4.1(2).

Finally, property (4) is clear by construction.

Lemma 2.1 describes what is needed to build F̄. Choose F̄(A_{i}) = B̂C(E_{i}), i = 1, …, 4. The maps F̄(f_{i}) (i = 1, 2, 3) are induced by the inclusions C(E_{i+1}) → C(E_{i}). The right actions of GL(2, F_{2}) and GL(3, F_{2}) on F̄(A_{2}) and F̄(A_{3}) respectively are obtained by composing the left actions of 6.1 with the inverse map. The right action of GL(4, F_{2}) on F̄(A_{4}) is obtained in a similar way from the natural left action of GL(4, F_{2}) on the based vector space E_{4} = C(E_{4}). By 6.1(3) the subgroup P (1, 1) ⊂ GL(2, F_{2}) acts on F̄(A_{2}) by automorphisms of C(E_{2}) which extend to inner automorphisms of Spin(7); this (cf. 3.6) gives relation 2.1(1). Relation 2.1(2) is 6.1(4). By 6.1(2), the restriction of the action of GL(3, F_{2}) on F̄(A_{3}) to P (1, 2) is induced by automorphisms of C(E_{3}) which extend to inner automorphisms of Spin(7); this establishes relation 2.1(4). Finally, relation 2.1(6) follows from 3.10, since the action of P (1, 3) on B̂C(E_{4}) is induced by group automorphisms which become inner in Spin(7).

Relations 2.1(3) and 2.1(5) require a little more work. For 2.1(3), let T_{ext}^{n}(2) denote the group of elements of order a power of 2 in T_{ext}^{n} (see 3.4), so that T_{ext}^{n}(2) is a semidirect product of (Z_{2^{∞}})^{n} with Z/2. The inclusion T_{ext}^{n}(2) → T_{ext}^{n} induces a cohomology isomorphism H^{∗}T_{ext}^{n} → H^{∗}T_{ext}^{n}(2) and therefore an equivalence B̂T_{ext}^{n}(2) ≅ B̂T_{ext}^{n}. Under this equivalence the above action of GL(3, F_{2}) on B̂C(E_{3}) ≅ B̂T_{ext}^{3} is by construction induced by an action of GL(3, F_{2}) on T_{ext}^{n}(2) by group homomorphisms. This last action is obtained by letting GL(3, F_{2}) act via ρ̂_{GL} on (Z_{2^{∞}})^{3} (note that Aut((Z_{2^{∞}})^{3}) ≅ GL(3, Ẑ_{2})) and then extending to T_{ext}^{3}(2) by using the fact that −I_{3} ∈ GL(3, Ẑ_{2}) commutes with ρ̂_{GL}(GL(3, F_{2})). In a similar way, the map F̄(f_{3}) : B̂E_{4} → B̂C(E_{3}) is induced by a group homomorphism E_{4} → T_{ext}^{3}(2) which sends the first three basis vectors of E_{4} to elements of order 2 in the subgroup (Z_{2^{∞}})^{3} and the last basis vector to an element of order 2 which does not lie in (Z_{2^{∞}})^{3}. Relation 2.1(3) now follows from a straightforward group theoretic calculation involving 6.1(1); note in particular that the automorphisms of E_{4} provided by U(3, 1) become inner automorphisms in T_{ext}^{3}(2) (or in C(E_{3})).

For 2.1(5) observe (3.10) that the automorphisms of E_{4} provided by P_{0}(2, 2) become inner automorphisms in C(E_{2}), so it is enough to check 2.1(5) for matrices h ∈ GL(4, F_{2}) which agree with the identity except in the upper left-hand 2 × 2 block. Such a matrix h belongs to P (3, 1), so that by 2.1(3) F̄(f_{3})F̄(h) = F̄(r_{3}(h)) and hence by 2.1(2) F̄(f_{2})F̄(f_{3})F̄(h) = F̄(r_{2}r_{3}(h))F̄(f_{2})F̄(f_{3}). This immediately gives 2.1(5) and completes the proof that F̄ is a functor A^{op} → Ho(T ).

Next we show that the functor H^{∗}F̄ is naturally equivalent to α_{R}. Let E_{A} : A → T be the functor which assigns to each object A_{i} the classifying space BE_{i}. A morphism f : A_{i} → A_{j} is sent by E_{A} to the map BE_{i} → BE_{j} induced by the homomorphism E_{i} → E_{j} corresponding to f under the unique basis-preserving isomorphisms A_{i} ≅ E_{i} and A_{j} ≅ E_{j}.

6.2 Lemma. The maps δ̂_{i} : BE_{i} → B̂C(E_{i}), i = 1, …, 4 (§3) are the components [26, p. 215] of a dinatural transformation δ̂: E_{A} →^{..} F̄, i.e., for each map f : A_{i} → A_{j} in A there is a commutative diagram

\begin{matrix}E_{A}(A_{i}) & ⟶^{δ̂_{i}} & F̄(A_{i}) \\ ^{E_{A}(f)}↓ & ^{F̄(f)}↑ \\ E_{A}(A_{j}) & ⟶^{δ̂_{j}} & F̄(A_{j}) \end{matrix} .

Proof: If f is an automorphism of A_{i} the above diagram commutes by 6.1(1), while if f is one of the basic maps f_{i} (2.1) it commutes by the definition of F̄(f_{i}). The lemma follows from the fact that these maps generate A.

Now let V be a rank four elementary abelian 2-group. Let M_{A} : A^{op} → Ho(T ) be the functor which assigns to A_{i} the function space Emb(BE_{i}, BV) ≅ Emb(E_{i}, V) × BV. As in the discussion preceding 3.13 there are canonical maps κ_{i,V} : M_{A}(E_{i}) → B̂C(E_{i}).

6.3 Lemma. The maps κ_{i,V}, i = 1, …, 4 are the components of a natural transformation κ_{V} : M_{A} → F̄.

Proof: We must show that for each morphism f : A_{i} → A_{j} in A and each g ∈ Emb(BE_{j}, BV) the following diagram in the homotopy category

Emb(BE_{i}, BV)_{g′} ⟶^{(κ_{i,V} )_{g′}} B̂C(E_{i})
(–)·E_{A}f↑  ^{F̄(f)}↑
Emb(BE_{j}, BV)_{g} ⟶_{(κ_{j,V} )_{g}} B̂C(E_{j})

commutes, where g′ = g · E_{A}(f) and subscripting a morphism denotes restriction to the indicated component of the domain. Each of the spaces on the left is equivalent by evaluation at the basepoint to BV, so the question is one of comparing two maps BV → B̂C(E_{i}). The upper map (κ_{i,V})_{g′} is the unique homotopy class which at the same time makes H^{∗}BV a finitely generated module over H^{∗}BC(E_{i}) and satisfies (κ_{i,V})_{g′} · g′ = δ̂_{i}. The other map, F̄(f) · (κ_{j,V})_{g}, is the composite with F̄_{f} of the unique homotopy class which makes H^{∗}BV a finitely generated module over H^{∗}B̂C(E_{j}) and satisfies (κ_{j,V})_{g} · g = δ̂_{j}. The two maps agree because F̄(f) · δ̂_{j} · E_{A}(f) = δ̂_{i} (see 6.2) and H^{∗}F̄(f) makes H^{∗}B̂C(E_{i}) a finitely generated module over H^{∗}B̂C(E_{j}). (This last follows from the fact that F̄(f) differs only by homotopy equivalences in the domain and range from a map induced by an inclusion K_{1} → K_{2} of compact Lie groups: any such map makes H^{∗}B̂K_{1} a finitely generated module over H^{∗}B̂K_{2} [29, 2.4].)

In the same way as in 3.13, the transformation κ_{V} is equivariant with respect to the natural action of Aut(V) on M_{A} and the trivial action of Aut(V) on F̄; it follows from 3.13 that the induced cohomology map κ_{V}^{∗} : H^{∗}F̄ → (H^{∗}M_{A})^{Aut(V)} is an isomorphism. The equivalence H^{∗}F̄ ≅ α_{R} follows from the fact that (H^{∗}M_{A})^{Aut(V)} is clearly up to equivalence the functor α_{R} of 2.3.

§7. Lifting to the category of spaces

In this section we will use the results of [6] to show that the functor F̄ : A^{op} → Ho(T ) can be lifted to a functor F : A^{op} → T . This proves 2.5 and hence 1.1.

Recall that a homotopy class of maps f : X → Y is said to be centric [6, §1] if precomposition with f induces a homotopy equivalence Aut(X)_{1} → Map(X, Y)_{f}. A functor Φ : C → Ho(T ) is centric if Φ(f) is a centric map for each morphism f of C.

7.1 Proposition. The functor F̄: A^{op} → Ho(T ) is centric.

This is [6, 4.2]. (Note that the partial completion functor which appears in the statement of [6, 4.2] agrees with the F_{2}-completion functor in the cases at hand because the fundamental groups of the spaces involved are finite 2-groups [3, II, 5.2(iv)].)

7.2 Lemma. The natural map (cf. [6, 4.5])

BE_{i} ≅ BZ(C(E_{i})) → Aut(B̂C(E_{i}))_{1}

is an equivalence for i = 2, …, 4.

Proof: The case i = 2 is 5.5. Cases i = 3 and i = 4 can be handled by explicit calculation (cf. [6, 5.3]).

Let γ_{j}F̄ (j ≥ 1) be the functor [6, §1] from A to the category of abelian groups with which assigns to A the group π_{j}Aut(F̄(A))_{1}; there is no basepoint problem with these homotopy groups because Aut(F̄(A))_{1} is a simple space. For each map f : A → A′ in A the induced map γ_{j}F̄(f) : γ_{j}F̄(A) → γ_{j}F̄(A′) is the composite

π_{j}Aut(F̄(A))_{1} ⟶^{h_{#}} π_{j}Map(F̄(A′), F̄(A))_{h} ⟶^{(h^{#})^{−1}} π_{j}Aut(F̄(A′))_{1}

where h is F̄(f), h_{#} is induced by postcomposition with h and h^{#} by precomposition.

Recall from §6 that E_{A} is a functor which sends the object A_{i} of A to BE_{i}. The natural action (7.2) of BE_{i} on B̂C(E_{i}) gives rise to maps π_{j}BE_{i} → π_{j}Aut(B̂C(E_{i}))_{1}.

7.3 Lemma. The above maps π_{j}BE_{i} → π_{j}Aut(B̂C(E_{i}))_{1} are the components of a natural transformation π_{j}E_{A} → γ_{j}F̄.

Proof: An examination of the definition of γ_{k}F̄ shows that by 7.2 it is enough to check that for each morphism f : A_{α} → A_{β} of A the following diagram in Ho(T ) commutes:

\begin{matrix}BE_{α} × B̂C(E_{β}) & ⟶^{1×F̄(f)} & BE_{α} × B̂C(E_{α}) \\ ^{E_{A}(f)×1}↓ & & ^{m}↓ \\ BE_{β} × B̂C(E_{β}) & ⟶^{F̄(f)·m} & B̂C(E_{α}) \end{matrix} .

Here m stands for the map BE_{i} × B̂C(E_{i}) → B̂C(E_{i}) induced by the group homomorphism E_{i} × C(E_{i}) → C(E_{i}). Restricting to the axes BE_{α} × pt and pt × B̂C(E_{β}) respectively gives the diagrams

\begin{matrix}BE_{α} & ⟶^{1} & BE_{α} \\ ^{E_{A}(f)}↓ & & ↓δ̂_{α} \\ BE_{β} & ⟶^{F̄(f)·δ̂_{β}} & B̂C(E_{α}) \end{matrix}  \begin{matrix}B̂C(E_{β}) & ⟶^{F̄(f)} & B̂C(E_{α}) \\ ^{1}↓ & & ^{1}↓ \\ B̂C(E_{β}) & ⟶^{F̄(f)} & B̂C(E_{α}) \end{matrix}

the first of which commutes by 6.2 and the second of which trivially commutes. The lemma follows from the fact that the basepoint evaluation map Map(BE_{α}, B̂C(E_{α}))δ̂_{α} → B̂C(E_{α}) is an equivalence ([14] and 5.3), so that a map BE_{α} × B̂C(E_{β}) → B̂C(E_{α}) which agrees with δ̂_{α} on BE_{α} × pt is actually determined up to homotopy by its restriction to pt × B̂C(E_{β}).

7.4 Lemma. The abelian groups \underleftarrow{lim}^{k} π_{j}E_{A} vanish for j, k > 0.

Proof: This is trivial if j ≠ 1, since in that case the functor π_{j}E_{A} itself vanishes. Let Γ = Aut(A_{4}) and let M be the Γ module A_{4} itself. Inspection then shows that the category A is equivalent to the category A_{Γ} of §8 in such a way that the functor π_{1}E_{A} corresponds to the functor α_{Γ,M} of 8.1. Case j = 1 of the lemma thus follows from 8.1.

7.5 Lemma. If γ is a functor from A to the category of abelian groups such that γ(A_{i}) = \{0\} for i ≠ 1, then \underleftarrow{lim}^{k} γ vanishes for k > 0.

Proof: This is an easy calculation with the standard cochain complex [3, XI, 6.2] for computing \underleftarrow{lim}^{∗} γ; in dimension k this cochain complex contains a product

∏_{a_{0}⟶a_{1}⟶···⟶a_{k}} γ(a_{k})

indexed by k-simplices of the nerve of A. Under the given circumstances this is the complex for computing the singular cohomology of a point with coefficients in γ(A_{0}).

7.6 Lemma. The abelian groups \underleftarrow{lim}^{k} γ_{j}F̄ vanish for j, k > 0.

Proof: By 7.2 the natural transformation of 7.3 is an isomorphism at the objects A_{i}, i ≠ 1, of A. It is easy to check that the natural transformation is a monomorphism at A_{1}; for j > 1 this is the case because the domain group is zero, while for j = 1 it follows from naturality and the fact that for any morphism A_{1} → A_{2} of A the induced map E_{1} → E_{2} is injective. If γ is the cokernel of this natural transformation, then by 7.5 the groups \underleftarrow{lim}^{k} γ vanish for k > 0. The lemma now follows from 7.4 and the long exact \underleftarrow{lim}^{∗} sequence [3, XI, 6.1] associated to a short exact sequence of functors.

Remark: Jackowski, McClure and Oliver have recently shown that 7.2 also holds for i = 1; given this result, which is difficult, 7.6 follows directly from 7.4.

7.7 Proposition. The functor F̄ : A^{op} → Ho(T ) can be lifted to a functor F : A^{op} → T .

Proof: Given 7.6, this is a consequence of the main theorem of [6]. In fact, the results of [6] show that up to a strong notion of equivalence there is only one lift F of F̄. Remark: Given a centric diagram F̄ in the homotopy category such that the realization obstruction groups [6] for F̄ vanish (7.6) and such that H^{∗}F̄ is equivalent to α_{R} (§2), Oliver has shown how to construct BDI(4) directly; this provides a proof of 1.1 which does not use 7.7.

§8. Derived functors of the inverse limit functor

In this section we will prove a certain vanishing theorem for the higher derived functors \underleftarrow{lim}^{i} [3, XI, §6] of the inverse limit functor. This vanishing theorem figures in the proof of 2.4 and in the proof of 7.4. We work in greater generality than we strictly need in order to emphasize the analogy with [13].

Let p be a prime number (not necessarily 2), V a finite-dimensional vector space over F_{p}, Γ ⊂ Aut(V) a subgroup and M a module over F_{p}[Γ]. For any F_{p} vector space A the group Γ acts by composition on Hom(A, V). Define A_{Γ} to be the category of pairs (A, O) where A is a non-zero F_{p} vector space and O is a Γ orbit of monomorphisms A → V; a map (A, O) → (A′, O′) is a monomorphism f : A → A′ such that g ∘ f ∈ O whenever g ∈ O′.

Let α_{Γ,M} be the functor from A_{Γ} to abelian groups which assigns to (A, O) the vector space of Γ-maps O → M. For any object x of A_{Γ} there is an evident map M^{Γ} → α_{Γ,M}(x); taken together these induce a map M^{Γ} → \underleftarrow{lim} α_{Γ,M}.

8.1 Proposition. The above map M^{Γ} → \underleftarrow{lim} α_{Γ,M} is an isomorphism.

The groups \underleftarrow{lim}^{i} α_{Γ,M} vanish for i ≥ 1.

Remark: Let α_{Γ,M}^{k} be the functor from A_{Γ} to abelian groups which assigns to (A, O) the group Ext_{F_{p}[Γ]}^{k}(F_{p}[O], M). (Note that α_{Γ,M}^{0} = α_{Γ,M} and that α_{Γ,M}^{k}(A, O) is isomorphic to H^{k}(Γ_{f}, M), where Γ_{f} ⊂ Γ is the isotropy subgroup of any f ∈ O.) The proof of 8.1 shows that \underleftarrow{lim} α_{Γ,M}^{k} ≅ H^{k}(Γ, M) and that \underleftarrow{lim}^{i} α_{Γ,M}^{k} = 0 for i ≥ 1.

8.2 Lemma. The conclusion of 8.1 is true under the additional assumption that Γ is a p-group.

Proof: (cf. [13, proof of 4.10]) Since Γ is a finite p-group, V^{Γ} ≠ {0}. Denote by F the object of A_{Γ} corresponding to the inclusion ι : V^{Γ} → V.

Let F ↓ A_{Γ} be the under category [26, p. 46] and j : F ↓ A_{Γ} → A_{Γ} the forgetful functor. The category F ↓ A_{Γ} has the identity map of F as an initial object and it is clear that the natural map M^{Γ} → α_{Γ,M}(F) =

α_{Γ,M}(j(F ⟶^{1} F)) is an isomorphism; it follows [3, XI, 7.2 and 9.2] that the natural map M^{Γ} → \underleftarrow{lim}(α_{Γ,M} · j) is an isomorphism and that

\underleftarrow{lim}^{i}(α_{Γ,M} · j) vanishes for i ≥ 1. Given an arbitrary object x = (A, O) of A_{Γ} choose f ∈ O and let σ(x) be the pair (A′, O′) where A′ is the quotient of A⊕V^{Γ} by the kernel of the sum map f +ι : A⊕V^{Γ} → V and O′ is the orbit of the monomorphism A′ → V. It is easy to check that σ(x) does not depend on the choice of orbit element f, that x ↦ σ(x) produces a functor σ : A_{Γ} → F ↓ A_{Γ}, and that the map A → A⊕V^{Γ} → A′ induces a natural transformation τ from the identity functor of A_{Γ} to the composite j · σ. By inspection the natural transformation from α_{Γ,M} to α_{Γ,M} · j · σ induced by τ is a natural equivalence (cf. 8.1).

Now let x be an object of F ↓ A_{Γ}, y an object of A_{Γ} and σ(y) → x a map of F ↓ A_{Γ}. It is clear that there is a unique map w : y → j(x) in A_{Γ} such that the given map σ(y) → x is the composite of σ(w) with the evident natural isomorphism σ(j(x)) ≅ x (in fact, σ is left adjoint to j). This shows that the over category σ ↓ x has a terminal object and therefore a contractible nerve [3, XI, §2]; consequently, the functor σ is left cofinal [3, XI, §9]. The proposition now follows directly from the fact [3, XI, 7.2 and 9.2] that the natural maps \underleftarrow{lim}^{i}(α_{Γ,M} ·

j) → \underleftarrow{lim}^{i}(α_{Γ,M} · j · σ) ≅ \underleftarrow{lim}^{i} α_{Γ,M} are isomorphisms for all i. (The

isomorphism \underleftarrow{lim}^{i}(α_{Γ,M} · j) ≅ \underleftarrow{lim}^{i} α_{Γ,M} is in fact a general consequence of the fact that j has a left adjoint [17]).

Let B be the category whose objects are non-trivial vector spaces of dimension ≤ dim(V) and whose morphisms are vector space monomorphisms. Let β_{Γ,M} be the functor from B to abelian groups which assigns to each object A the vector space of Γ-maps Emb(A, V) → M (here as in §5 Emb(A, V) is the set of monomorphisms A → V). The unique Γ-map Emb(A, V) → ∗ induces for each A a map M^{Γ} → β_{Γ,M}(A); these combine to give a map M^{Γ} → \underleftarrow{lim} β_{Γ,M}.

8.3 Lemma. There are natural isomorphisms \underleftarrow{lim}^{i} α_{Γ,M} → \underleftarrow{lim}^{i} β_{Γ,M} for all i ≥ 0. Under these isomorphisms the natural map M^{Γ} → \underleftarrow{lim} α_{Γ,M} corresponds to the natural map M^{Γ} → \underleftarrow{lim} β_{Γ,M}.

The proof of this is identical to the proof of [13, 2.3]. The standard cochain complexes which compute the two higher limits involved are isomorphic.

Proof of 8.1: (cf. [13, proof of 1.2]) By 8.3 it is enough to prove that the natural map M^{Γ} → \underleftarrow{lim} β_{Γ,M} is an isomorphism and that \underleftarrow{lim}^{i} β_{Γ,M} vanishes for i ≥ 1. Let Π ⊂ Γ be a p-Sylow subgroup. Since the index of Π in Γ is prime to p, the transfer construction [4, p. 80] exhibits β_{Γ,M} as a natural retract of β_{Π,M}. This implies that \underleftarrow{lim}^{i} β_{Γ,M} is a retract

of \underleftarrow{lim}^{i} β_{Π,M} (i ≥ 0). The desired result now follows from 8.2, since a retract of an isomorphism is an isomorphism (i = 0) and a retract of {0} is {0} (i ≥ 1).

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