Gysin_V-functors

Abstract

Let $d\ge 1$ be an integer and ${\mathcal{K}}_d$ be a contravariant functor from the category of subgroups of $(\mathbb{Z}/2\mathbb{Z}{)}^d$ to the category of graded and finite ${\mathbb{F}}_2$-algebras. In this paper, we generalize the conjecture of G. Carlsson, concerning free actions of $(\mathbb{Z}/2\mathbb{Z}{)}^d$ on finite CW-complexes, by suggesting, that if ${\mathcal{K}}_d$ is a Gysin-$(\mathbb{Z}/2\mathbb{Z}{)}^d$-functor (that is to say, the functor ${\mathcal{K}}_d$ satisfies some properties), then we have: $\left(C_d\right):\sum _{i\ge 0}di{m}_{{\mathbb{F}}_2}({\mathcal{K}}_d(0){)}^i\ge 2^d$.\\ We prove this conjecture for $1\le d\le 3$ and we show that, in certain cases, we get an independent proof of the following result.\\ Theorem. If the group $(\mathbb{Z}/2\mathbb{Z}{)}^d$, $1\le d\le 3$, acts freely and cellularly on a finite CW-complex $X$, then $\sum _{i\ge 0}di{m}_{{\mathbb{F}}_2}{H}^i(X;{\mathbb{F}}_2)\ge 2^d$.

16 pages