94d:55033 55R35 22E99

Mislin, Guido ^{(}1-OHS^{)}

Lannes’ T -functor and the cohomology of BG. Compositio Math. 86 (1993), no. 2, 177–187. In this paper it is shown that compact Lie groups are “characterized” up to isomorphism by the high-dimensional cohomology of their classifying spaces. The approach is as follows. Define a category 𝒜_{R} with objects the finite morphisms R → H^{∗}A, where R is an object of the category 𝒦 of unstable algebras over the mod-p Steenrod algebra, A is a nontrivial elementary abelian p-group and H^{∗}A = H^{∗}(A; Z/p). Let the functor α_{R}: 𝒜_{R} → 𝒦 be defined by taking f: R → H^{∗}A to T_{f}R = T_{A}R ⊗_{(T_{A}R)^{0}} F_{p}, where T_{A} is Lannes’ T -functor, and let α(R) ∈ 𝒦 denote the inverse limit of the α_{R}. The author shows that, for R connected and finitely generated as a ring, α(R) ≅ α(R⟨n⟩), where (R⟨n⟩)^{0} = R^{0}, (R⟨n⟩)^{j} = R^{j} for j ≥ n and (R⟨n⟩)^{j} = 0 otherwise. Letting R = H^{∗}(BG; Z/p) and combining the discussion above with the isomorphism H^{∗}(BG; Z/p) →^{≅} α(H^{∗}(BG; Z/p)) of W. G. Dwyer and C. W. Wilkerson [Topology 31 (1992), no. 2, 433–443; MR 93h:55008], the author obtains the “reconstruction” isomorphism α(H^{∗}(BG; Z/p)⟨n⟩) ≅ H^{∗}(BG; Z/p), which characterizes H^{∗}(BG; Z/p) in terms of its high-dimensional behavior. One application of this is the following: If ρ: G → H is a homomorphism of compact Lie groups which induces isomorphisms (ρ^{∗})^{j}: H^{j}(BH; Z) → H^{j}(BG; Z) for all j ≥ n (some n), then, in fact, (ρ^{∗})^{j} is an isomorphism for all j ≥ 0. By results of S. Jackowski [Topology 17 (1978), no. 3, 303–307; MR 81j:20071] and N. Minami [Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 81–93; MR 89h:55007], this then implies that ρ: G → H is an isomorphism of Lie groups. John F. Oprea (1-CVLS)

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