Gysin_V-functors

TL;DR

This paper generalizes a conjecture related to free actions of elementary abelian 2-groups on finite CW-complexes, proving it for dimensions 1 to 3 and deriving lower bounds on cohomology dimensions.

Contribution

It introduces Gysin-$(bZ/2bZ)^d$-functors and proves a conjecture for dimensions up to 3, providing new proofs of known results about free group actions.

Findings

Proved the conjecture for 1 ≤ d ≤ 3.

Established lower bounds on the sum of Betti numbers for free actions.

Provided an independent proof of a known theorem for small d.

Abstract

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 ${\underset{i \geq 0}{\sum}}dim_{\mathbb{F}_{2}}H^{i}(X;\; \mathbb{F}_{2}) \geq 2^{d}$.