On a Relative Hilali Conjecture

Abstract

The well-known Hilali conjecture stated in [9] is one claiming that if $X$ is a simply connected elliptic space, then $ \dim \pi_*(X)\otimes {\mathbb Q} \leq \dim H_*(X; {\mathbb Q})$. In this paper we propose that if $f:X \to Y$ is a continuous map of simply connected elliptic spaces, then $\dim {\rm Ker} \ \pi_*(f)_{\mathbb Q}\leq \dim {\rm Ker}\ H_*(f; {\mathbb Q})+1$, and we prove this for certain reasonable cases. Our proposal is a relative version of the Hilali conjecture and it includes the Hilali conjecture as a special case.