Sur la cohomologie modulo p des p-groupes abeliens élémentaires. (On the cohomology modulo p of elementary Abelian p-groups). (French) Zbl 0654.55013

Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 97-116 (1987).
[For the entire collection see Zbl 0628.00011.]
Let A be the Steenrod algebra modulo p and \({\mathfrak U}\) the category of unstable A-modules. Put \(H^*(V)=H^*(BV,F_ p)\) for an elementary Abelian p-group V. The author proves that the functor \(T_ V: {\mathfrak U}\to {\mathfrak U}\), the left adjoint to the functor: \({\mathfrak U}\to {\mathfrak U}\), \(N\mapsto H^*(V)\otimes N\) is exact and preserves the tensor product [cf. the author and S. Zarati, Ann. Sci. Ec. Norm. Sup., IV. Ser. 19, 303-333 (1986; Zbl 0608.18006)]. Then some homotopical applications are given. In particular, the set of homotopy classes [BV,Y], for special spaces Y is described to answer in the affirmative a conjecture of H. R. Miller [Lect. Notes Math. 1051, 401-417 (1984; Zbl 0542.55016)] and a generalized Sullivan conjecture [cf. loc. cit.].
Reviewer: M.Golasiński


55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55S10 Steenrod algebra
55P99 Homotopy theory
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)