## 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

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)].

### MSC:

 55P99 Homotopy theory 55S10 Steenrod algebra

Zbl 0654.55013
Full Text:

### References:

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