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

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 $r\left(P(t),Q(t);\epsilon \right)$ 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 $t\ge \epsilon$. This range is related to the classical Lambert W-function $W(z)$.

any comments are welcome, to appear in the "Proceedings of the 2019 Hyper-JARCS Conference"

References in corpus (2)

• General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function
• Classification of rationally elliptic toric orbifolds