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

TL;DR

This paper investigates conditions under which the loop space of sphere bundle total spaces over highly connected Poincaré complexes splits into a product of simpler loop spaces, advancing understanding of their topological structure.

Contribution

It establishes new sufficient conditions for loop space splittings of sphere bundles over highly connected Poincaré complexes.

Findings

Identifies conditions for loop space splitting as a product of loop spaces of base and fiber.

Provides a framework for analyzing the topology of sphere bundle total spaces.

Enhances understanding of the homotopy type of sphere bundles over complex spaces.

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 $S^{m-1}$.