# Gysin-(ℤ/2ℤ)d-functors

## Les foncteurs de type Gysin-(ℤ/2ℤ)d

Dorra Bourguiba
Université Tunis-ElManar
Tunisie

Said Zarati
Université Tunis-ElManar
Tunisie

Published on 7 March 2023   DOI : 10.21494/ISTE.OP.2023.0939

### Mots-clés

Let d ≥ 1 be an integer and Kd 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 Kd is a Gysin-(ℤ/2ℤ)d-functor (that is to say, the functor Kd 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 Kd 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 Kd is a Gysin-(ℤ/2ℤ)d-functor (that is to say, the functor Kd 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 ${\underset{i \geq 0}{\sum}}dim_{\mathbb{F}_{2}}H^{i}(X;\; \mathbb{F}_{2}) \geq 2^{d}$