Poincaré polynomials of a map and a relative Hilali conjecture

T. Yamaguchi, S. Yokura

In this paper we introduce homological and homotopical Poincaré polynomials Pf(t) and Pπf(t) of a continuous map f : X → 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 PX(t) and PπX(t) of the source space X. Our relative Hilali conjecture Pπf(1) ≤ Pf(1) is a map version of the the well-known Hilali conjecture PπX(1) ≤ PX(1) of a rationally elliptic space X. In this paper we show that under the condition that Hi(f; Q) : Hi(X; Q) → Hi(Y; Q) is not injective for some i>0, the relative Hilali conjecture of product of maps holds, namely, there exists a positive integer n0 such that for all n ≤ n0 the strict inequality Pπfn(1) < Pfn(1) holds. In the final section we pose a question whether a Hilali-type inequality HPπX(rX) ≤ P X(rX) holds for a rationally hyperbolic space X, provided the the homotopical Hilbert-Poincare series HPπX(rX) converges at the radius rX of convergence. 

Tbilisi Mathematical Journal, Vol. 13(4) (2020), pp. 33-47