Homology, Homotopy and Applications

Volume 24 (2022)

Number 2

Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

Pages: 93 – 113

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a5


Anatoly Libgober (Department of Mathematics, University of Illinois, Chicago, Il., U.S.A.)

Shoji Yokura (Graduate School of Science and Engineering, Kagoshima University, Kagoshima, Japan)


We discuss inequalities between the values of homotopical and cohomological Poincaré polynomials of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called stabilization threshold $\mathfrak{pp}(X; \varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $\varepsilon$, and we show that the Hilali conjecture implies that $\mathfrak{pp}(X; 1) \leqslant 3$.


mixed Hodge structure, mixed Hodge polynomial, Hilali conjecture, rational homotopy theory

2010 Mathematics Subject Classification

32S35, 55N99, 55P62, 55Q40

Received 9 November 2020

Received revised 11 June 2021

Accepted 22 July 2021

Published 10 August 2022