Self-products of rationally elliptic spaces and inequalities between the ranks of homotopy and homology groups

TL;DR

This paper surveys recent inequalities relating homotopy and homology ranks of rationally elliptic spaces and introduces a new invariant to compare these ranks, connecting algebraic topology with polynomial inequalities and the Lambert W-function.

Contribution

It introduces a novel invariant for rationally elliptic spaces that refines existing inequalities and links algebraic topology with polynomial inequalities involving the Lambert W-function.

Findings

New invariant for rationally elliptic spaces

Refined inequalities between homotopy and homology ranks

Connection to polynomial inequalities and Lambert W-function

Abstract

We give a survey on recent results on inequalities between the ranks of homotopy and cohomology groups (resp., graded components of mixed Hodge structures on these groups) of rationally elliptic spaces (resp., quasi-projective varieties which are rationally elliptic). We also discuss a refinement of these results describing a new invariant of rationally elliptic spaces allowing to compare the ranks of homotopy and homology groups. This invariant is a specialization of an invariant $r\left ( P(t),Q(t);\varepsilon \right )$ of a pair $\left (P(t),Q(t) \right )$ of polynomials with non-negative integer coefficients, describing the range of variable $r$ such that $rP(t)<Q(t)^r$ for all $t \ge \varepsilon$. This range is related to the classical Lambert W-function $W(z)$.