  Let $d \geq 1$ 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:
$\big(C_{d} \big): \; \underset{i \geq 0}{\sum}dim_{\mathbb{F}_{2}}
\big(\mathcal{K}_{d}(0)\big)^{i} \geq 2^{d}$.\\ We prove this conjecture for $1
\leq d \leq 3$ 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 \leq d \leq 3$, acts freely and cellularly
on a finite CW-complex $X$, then ${\underset{i \geq
0}{\sum}}dim_{\mathbb{F}_{2}}H^{i}(X;\; \mathbb{F}_{2}) \geq 2^{d}$.
