Poincaré polynomials of a map and a relative Hilali conjecture

Abstract

In this paper we introduce homological and homotopical Poincaré polynomials $P_f(t)$ and $P_f^{\pi }(t)$ 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 Poincaré polynomials are respectively equal to the usual homological and homotopical Poincaré polynomials $P_X(t)$ and $P_X^{\pi }(t)$ of the source space $X$. Our relative Hilali conjecture $P_f^{\pi }(1)\leqq P_f(1)$ is a map version of the the well-known Hilali conjecture $P_X^{\pi }(1)\leqq P_X(1)$ 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 $P_{f^n}^{\pi }(1)<P_{f^n}(1)$} holds, where $f^n:X^n\to Y^n$. In the final section we pose a question whether a "Hilali"-type inequality $H{P}_X^{\pi }(r_X)\leqq P_X(r_X)$ holds for a rationally hyperbolic space $X$, provided the the homotopical Hilbert--Poincare series $H{P}_X^{\pi }(r_X)$ converges at the radius $r_X$ of convergence.

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