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

Abstract

We survey recent results on inequalities between the ranks of homotopy and cohomology groups (resp., graded components of mixed Hodge structures on these groups) for 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. Its definition is based of the study of the range of parameter r such that rP(t)<Q(t)r for all t≥ε, where (P(t),Q(t)) is a pair of arbitrary polynomials with non-negative integer coefficients. We show that this range is related to the classical Lambert W-function W(z).