Poincar\'e polynomials of a map and a relative Hilali conjecture

TL;DR

This paper introduces homological and homotopical Poincaré polynomials for maps, proposes a relative Hilali conjecture, and proves a strict inequality for iterated maps under certain conditions, advancing the understanding of rational homotopy theory.

Contribution

It defines Poincaré polynomials for maps, formulates a relative Hilali conjecture, and proves a strict inequality for iterated maps when certain homology conditions are met.

Findings

Established a condition under which the relative Hilali conjecture holds for iterated maps.

Proved that for large n, the strict inequality between Poincaré polynomials of iterated maps is valid.

Posed an open question about a Hilali-type inequality for rationally hyperbolic spaces.

Abstract

In this paper we introduce homological and homotopical Poincar\'e polynomials $P_f(t)$ and $P^{\pi}_f(t)$ of a continuous map $f:X \to Y$ such that if $f:X \to Y$ is a constant map, or more generally, if $Y$ is contractible, then these Poincar\'e polynomials are respectively equal to the usual homological and homotopical Poincar\'e 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 \emph{strict inequality $P^{\pi}_{f^n}(1) < P_{f^n}(1)$} holds, where $f^n:X^n \to Y^n$. 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--Poincare series $HP^{\pi}_X(r_X)$ converges at the radius $r_X$ of convergence.