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 $P^{\pi}_f(t)$ 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.