Loop Space Splittings of Sphere Bundles over Highly Connected Poincar\'e Complexes

Abstract

Let $m > n \ge 2$, and let $N$ be an $(n-1)$-connected $2n$-Poincar\'e complex. In this paper, we establish sufficient conditions under which the loop space of the total space $M$ of the sphere bundle $S^{m-1} \to M \to N$ (associated to a rank-$m$ real vector bundle over $N$) splits as a product of the loop spaces of $N$ and $S^{m-1}$.