The Hilali conjecture on product of spaces

TL;DR

This paper investigates the Hilali conjecture for simply connected rationally elliptic spaces, demonstrating that for sufficiently large products, the strict inequality between rational homotopy and homology dimensions holds.

Contribution

It proves that for any such space, there exists an n₀ such that for all n ≥ n₀, the product space Xⁿ satisfies the strict inequality in the Hilali conjecture.

Findings

Existence of n₀ such that the inequality holds for all n ≥ n₀

The strict inequality is achieved in large product spaces

Supports the Hilali conjecture in the context of product spaces

Abstract

The Hilali conjecture claims that a simply connected rationally elliptic space $X$ satisfies the inequality $\operatorname{dim} (\pi_*(X)\otimes \mathbb Q ) \leqq \operatorname{dim} H_*(X;\mathbb Q )$. In this paper we show that for any such space $X$ there exists a positive integer $n_0$ such that for any $n \geqq n_0$ the \emph{strict inequality} $\operatorname{dim} (\pi_*(X^n)\otimes \mathbb Q ) < \operatorname{dim} H_*(X^n;\mathbb Q )$ holds, where $X^{n}$ is the product of $n$ copies of $X$.