On function spaces whose source is the classifying space of an elementary abelian \(p\)-group. (Sur les espaces fonctionnels dont la source est le classifiant d’un \(p\)-group abélien élémentaire. (Appendice par Michel Zisman).) (French) Zbl 0857.55011

Lannes, Jean

Publ. Math., Inst. Hautes Étud. Sci. 75, 135-244 (1992).

In this long paper, the author studies function spaces \(\text{hom} (BV,Y)\), consisting of all continuous mappings from \(BV\) to a topological space \(Y\), where \(BV\) denotes the classifying space of an elementary abelian \(p\)-group \(V\) for a fixed prime \(p\). His success in this project depends on several remarkable properties of \(H^* (BV)\), the \(\text{mod }p\) cohomology of \(BV\), as both unstable module and unstable algebra over the \(\text{mod }p\) Steenrod algebra \({\mathcal A}\). To state these properties, let \({\mathcal U}\) denote the category of unstable \({\mathcal A}\)-modules, and let \({\mathcal K}\) denote the category of unstable \({\mathcal A}\)-algebras. If \({\mathcal T}\) denotes either of these categories, and \(K\) is an object of \({\mathcal T}\) having finite dimension in each degree, then the functor \({\mathcal T}\to {\mathcal T}\), \(N\mapsto K\otimes N\), admits a left adjoint denoted by \(M\mapsto(M:K)_{\mathcal T}\). (One can view such functors as analogous to functors \(Y\mapsto\text{hom}(X,Y)\) in the category of spaces.) For example, if \(H^*X\) is finite dimensional in each degree, then there is a natural map \((H^*Y:H^*X)_{\mathcal K}\to H^*\text{hom}(X,Y)\); a main result of this paper asserts that this is often an isomorphism in the category \({\mathcal K}\) when \(X=BV\).
For an elementary abelian \(p\)-group, \(T_V\) denotes the functor \({\mathcal U}\to {\mathcal U}\), \(M\mapsto(M:H^*(BV))_{\mathcal U}\). It is proved that (a) the functor \(T_V\) is exact; (b) the functor \(T_V\) commutes with tensor products; and (c) if \(M\) is an unstable \({\mathcal A}\)-algebra, then \(T_VM\) can be given the structure of an unstable \({\mathcal A}\)-algebra so that \(T_VM\) coincides with \((M:H^*(BV))_{\mathcal K}\) in the category \({\mathcal K}\). The property (a) generalizes the result of Gunnar Carlsson and Haynes Miller that \(H^*(B\mathbb{Z}/p)\) is injective in \({\mathcal U}\).
The remaining task is to deduce properties of \(\text{hom} (BV,Y)\) and the set of homotopy classes \([BV,Y]\) from these algebraic properties of \(H^*(BV)\). For this, it is sometimes helpful to replace \(Y\) by its Bousfield-Kan \(p\)-completion \(\widehat Y\). One notable result asserts that the natural map \([BV,Y]\to\operatorname{Hom}_{\mathcal K} (H^*Y, H^*BV)\) is a bijection provided that \(Y\) is simply connected and \(H^*Y\) is finite dimensional in each degree. The principal result of the paper asserts that the natural map \(T_VH^*Y\to H^*\text{hom}(BV,\widehat Y)\) is an isomorphism of unstable \({\mathcal A}\)-algebras provided that \(H^*Y\) and \(T_VH^*Y\) are finite dimensional in each degree and \(T_VH^*Y\) is zero in degree 1.
A final chapter concerns homotopy fixed point spaces for actions of elementary abelian \(p\)-groups.
A number of the results of this paper were announced by the author in his paper [Homotopy theory, Proc. Symp. Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 97-116 (1987; Zbl 0654.55013)]. An excellent account of the context in which the results of this paper play a major clarifying rôle is given by Lionel Schwartz in his book [Unstable modules over the Steenrod algebra and Sullivan’s fixed point conjecture, Chicago Lectures in Mathematics, Chicago, IL: Univ. of Chicago Press, (1994)].

Reviewer: P.Landweber (New Brunswick)

MSC:

55P99	Homotopy theory
55S10	Steenrod algebra

Keywords:

Lannes’ \(T\)-functor; elementary abelian \(p\)-group; Bousfield-Kan \(p\)-completion; function spaces; classifying space; unstable module; unstable algebra; Steenrod algebra

Citations:

Zbl 0654.55013

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