Analytic functors, unstable algebras and cohomology of classifying spaces.

*(English)*Zbl 0683.55013
Algebraic topology, Proc. Int. Conf., Evanston/IL 1988, Contemp. Math. 96, 197-220 (1989).

[For the entire collection see Zbl 0673.00017.]

Let \({\mathcal K}\) denote the category of unstable modules over the mod p Steenrod algebra A, and \({\mathcal G}\) the category of covariant functors from the category of finite dimensional \(F_ p\)-vector spaces to the category of profinite sets. There is a functor g from \({\mathcal K}\) to \({\mathcal G}\) defined by \(g(K)(V)=Hom_{{\mathcal K}}(K,H^*V)\), where \(H^*V\) denotes the mod p cohomology of the elementary abelian group V. This functor factors faithfully through the category \({\mathcal K}/{\mathcal N}{\mathcal I}{\mathcal L}\) obtained from \({\mathcal K}\) by inverting morphisms whose underlying module morphism has nilpotent kernel and cokernel, and has image consisting precisely of the analytic functors, a notion which is defined in the paper. Only a brief sketch of the proof is presented, with details promised to follow in another paper. Several applications of this theorem are given. One is a slick proof of the embedding theorem of J. F. Adams and C. W. Wilkerson [Ann. Math., II. Ser. 111, 95-143 (1980; Zbl 0417.55018)]. Another is that any A-subalgebra of \((H^*V)^ n\) whose underlying A-module is \({\mathcal U}\)-injective can be realized as the cohomology algebra of a space. Finally, a number of known results about cohomology of classifying spaces BG of finite or compact Lie groups G are interpreted as results about the sets Rep(V,G) of G-conjugacy classes of morphisms from V to G.

Let \({\mathcal K}\) denote the category of unstable modules over the mod p Steenrod algebra A, and \({\mathcal G}\) the category of covariant functors from the category of finite dimensional \(F_ p\)-vector spaces to the category of profinite sets. There is a functor g from \({\mathcal K}\) to \({\mathcal G}\) defined by \(g(K)(V)=Hom_{{\mathcal K}}(K,H^*V)\), where \(H^*V\) denotes the mod p cohomology of the elementary abelian group V. This functor factors faithfully through the category \({\mathcal K}/{\mathcal N}{\mathcal I}{\mathcal L}\) obtained from \({\mathcal K}\) by inverting morphisms whose underlying module morphism has nilpotent kernel and cokernel, and has image consisting precisely of the analytic functors, a notion which is defined in the paper. Only a brief sketch of the proof is presented, with details promised to follow in another paper. Several applications of this theorem are given. One is a slick proof of the embedding theorem of J. F. Adams and C. W. Wilkerson [Ann. Math., II. Ser. 111, 95-143 (1980; Zbl 0417.55018)]. Another is that any A-subalgebra of \((H^*V)^ n\) whose underlying A-module is \({\mathcal U}\)-injective can be realized as the cohomology algebra of a space. Finally, a number of known results about cohomology of classifying spaces BG of finite or compact Lie groups G are interpreted as results about the sets Rep(V,G) of G-conjugacy classes of morphisms from V to G.

Reviewer: D.M.Davis

##### MSC:

55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |