Complexes de modules \'equivariants sur l'alg\`ebre de Steenrod associ\'es \`a un $(\mathbb{Z}/2)^{n}$-CW-complexe fini

TL;DR

This paper investigates two cochain complexes over the mod2 Steenrod algebra associated with a finite elementary abelian 2-group action on a CW-complex, using both topological and algebraic approaches rooted in unstable modules.

Contribution

It introduces and compares topological and algebraic complexes for equivariant cohomology, utilizing unstable module theory and offering new techniques distinct from prior work.

Findings

Establishes a relationship between the topological and algebraic complexes.

Develops new methods based on unstable module theory.

Provides insights into equivariant cohomology for finite elementary abelian 2-groups.

Abstract

Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of $\mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the "topological complex", is defined using the orbit filtration of $X$. The second, which we call the "algebraic complex", is defined just in terms of the unstable $\mathrm{A}$-module $\mathrm{H}^*_V X$, the mod2 equivariant cohomology of $X$. Our study makes intensive use of the theory of unstable $\mathrm{H}^{*}V$-$\mathrm{A}$-modules which is a by-product of the researches on Sullivan conjecture. There is a noteworthy overlap between the topological part of our memoir and the paper "Syzygies in equivariant cohomology in positive characteristic", by Allday, Franz and Puppe, which has just appeared; however our techniques are quite different from theirs (the name "Steenrod" does not show up in their article).