On the homology of Postnikov fibres

Abstract:

Let $k$ be a field of positive characteristic and $X$ be a simply connected space of the homotopy type of a finite type CW complex. The Postnikov fibre ${X_{[n]}}$ of $X$ is defined as the homotopy fibre of the $n$-equivalence ${f_n}:X \to {X_n}$ coming from the Postnikov tower $\{ {X_n}\}$ of $X$. We prove that if the Lusternik-Schnirelmann category of $X$ is finite, then ${H_{\ast }}({X_{[n]}};k)$ contains a free module on a subalgebra $K$ of ${H_{\ast }}(\Omega {X_n};k)$ such that ${H_{\ast }}(\Omega {X_n};k)$ is a finite-dimensional free $K$-module.