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December 2020 Poincaré polynomials of a map and a relative Hilali conjecture
Toshihiro Yamaguchi, Shoji Yokura
Tbilisi Math. J. 13(4): 33-47 (December 2020). DOI: 10.32513/tbilisi/1608606048

Abstract

In this paper we introduce homological and homotopical Poincaré polynomials $P_f(t)$ and $P^{\pi}_f(t)$ of a continuous map $f:X \to Y$ such that if $f$ is a constant map, or more generally, if $Y$ is contractible, then these Poincaré polynomials are respectively equal to the usual homological and homotopical Poincaré polynomials $P_X(t)$ and $P^{\pi}_X(t)$ of the source space $X$. Our relative Hilali conjecture $P^{\pi}_f(1) \leqq P_f(1)$ is a map version of the the well-known Hilali conjecture $P^{\pi}_X(1) \leqq P_X(1)$ of a rationally elliptic space X. In this paper we show that under the condition that $H_i(f;\mathbb Q):H_i(X;\mathbb Q) \to H_i(Y;\mathbb Q)$ is not injective for some $i>0$, the relative Hilali conjecture of product of maps holds, namely, there exists a positive integer $n_0$ such that for $\forall n \geqq n_0$ the strict inequality $P^{\pi}_{f^n}(1) \leq P_{f^n}(1)$ holds. In the final section we pose a question whether a "Hilali"-type inequality $HP^{\pi}_X(r_X) \leqq P_X(r_X)$ holds for a rationally hyperbolic space $X$, provided the the homotopical Hilbert-Poincaré series $HP^{\pi}_X(r_X)$ converges at the radius $r_X$ of convergence.

Funding Statement

T.Y. is supported by JSPS KAKENHI Grant Number JP20K03591 and S.Y. is supported by JSPS KAKENHI Grant Number JP19K03468.

Acknowledgment

We would like to thank the referee for his/her thorough reading and useful comments and suggestions.

Citation

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Toshihiro Yamaguchi. Shoji Yokura. "Poincaré polynomials of a map and a relative Hilali conjecture." Tbilisi Math. J. 13 (4) 33 - 47, December 2020. https://doi.org/10.32513/tbilisi/1608606048

Information

Received: 11 May 2020; Accepted: 18 September 2020; Published: December 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4194227
Digital Object Identifier: 10.32513/tbilisi/1608606048

Subjects:
Primary: 06A06
Secondary: 18B35, 18B99, 54B99, 55N99, 55P62, 55P99

Rights: Copyright © 2020 Tbilisi Centre for Mathematical Sciences

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Vol.13 • No. 4 • December 2020
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