Gysin-(ℤ/2ℤ)^d-functors

Let d ≥ 1 be an integer and K_d be a contravariant functor from the category of subgroups of (ℤ/2ℤ)^d to the category of graded and finite 𝔽₂-algebras. In this paper, we generalize the conjecture of G. Carlsson [C3], concerning free actions of (ℤ/2ℤ)^d on finite CW-complexes, by suggesting, that if K_d is a Gysin-(ℤ/2ℤ)^d-functor (that is to say, the functor K_d satisfies some properties, see 2.2), 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 ≤ d ≤ 3 and we show that, in certain cases, we get an independent proof of the following
results (for d = 3 see [C4]):
If the group (ℤ/2ℤ)^d, 1 ≤ d ≤ 3, acts freely and cellularly on a finite CW-complex X, then $$${\underset{i \geq 0}{\sum}}dim_{\mathbb{F}_{2}}H^{i}(X;\; \mathbb{F}_{2}) \geq 2^{d}$$$

Let d ≥ 1 be an integer and K_d be a contravariant functor from the category of subgroups of (ℤ/2ℤ)^d to the category of graded and finite 𝔽₂-algebras. In this paper, we generalize the conjecture of G. Carlsson [C3], concerning free actions of (ℤ/2ℤ)^d on finite CW-complexes, by suggesting, that if K_d is a Gysin-(ℤ/2ℤ)^d-functor (that is to say, the functor K_d satisfies some properties, see 2.2), 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 ≤ d ≤ 3 and we show that, in certain cases, we get an independent proof of the following
results (for d = 3 see [C4]) :
If the group (ℤ/2ℤ)^d, 1 ≤ d ≤ 3, acts freely and cellularly on a finite CW-complex X, then $$${\underset{i \geq 0}{\sum}}dim_{\mathbb{F}_{2}}H^{i}(X;\; \mathbb{F}_{2}) \geq 2^{d}$$$

Elementary abelian 2-groups H∗(ℤ/2ℤ)^d-modules [en] H∗(ℤ/2ℤ)^d− Free actions of (ℤ/2ℤ)^d on finite CW-complexes Equivariant cohomology Gysin exact sequence

Elementary abelian 2-groups H∗(ℤ/2ℤ)^d-modules [fr] H∗(ℤ/2ℤ)^d− Free actions of (ℤ/2ℤ)^d on finite CW-complexes Equivariant cohomology Gysin exact sequence