A Cohomology Decomposition Theorem

In [9] Jackowski and McClure gave a homotopy decomposition theorem for the classifying space of a compact Lie group G; their theorem states that for any prime p the space BG can be constructed at p as the homotopy direct limit of a specific diagram involving the classifying spaces of centralizers of elementary abelian p-subgroups of G. In this paper we will prove a parallel algebraic decomposition theorem for certain kinds of unstable algebras over the mod p Steenrod algebra. This algebraic result gives a new proof of the theorem of Jackowski and McClure and has the potential to lead to homotopy decompositon theorems for many spaces which are not of the form BG (see §6). Before stating our results we will recall some material from [9]. Choose a prime p. Let G be a compact Lie group, and let AG be the category whose objects are the non-trivial elementary abelian p-subgroups of G; a morphism A → A′ in AG is a homomorphism f : A → A′ of abelian groups with the property that there exists an element g ∈ G such that f(x) = gxg−1 for all x ∈ A. There is a functor from A G to the category of topological spaces which sends A to the Borel construction EG ×G (G/C(A)), where C(A) denotes the centralizer of A in G. (Notice that this Borel construction has the homotopy type of the classifying space BC(A).) Jackowski and McClure prove that the natural map from the homotopy direct limit of this functor to EG×G ∗ = BG is an isomorphism on mod p cohomology. They derive this from a spectral sequence argument [2, XII, 5.8] and the following calculation with the inverse limit functor lim ← and its right derived functors lim ← . Let H∗ denote mod p cohomology and αG the functor on AG which sends A to H∗(EG×G (G/C(A)). Theorem 1.1 [9, Prop. 3–4]. The natural map H∗BG → lim ← αG is an isomorphism and the groups lim ← αG vanish for i > 0. The proof of Theorem 1.1 in [9] uses the Feshbach double coset formula and so depends heavily on the presence of a genuine compact Lie group.