Sphere bundles over $4$-manifolds are trivial after looping

TL;DR

This paper proves that, with two exceptions, sphere bundles over simply connected 4-manifolds become trivial after looping, implying their total spaces are homotopy equivalent in loop space despite being inequivalent as bundles.

Contribution

It establishes that most sphere bundles over 4-manifolds trivialize after looping, revealing a homotopy equivalence among their total spaces' loop spaces.

Findings

Sphere bundles over 4-manifolds trivialize after looping (except two cases)

Loop spaces of total manifolds are homotopy equivalent despite bundle inequivalence

Infinite inequivalent sphere bundles have homotopy equivalent loop space structures

Abstract

We show that except two special cases, the sphere bundle of a vector bundle over a simply connected $4$-manifold splits after looping. In particular, this implies that though there are infinitely many inequivalent sphere bundles of a given rank over a $4$-manifold, the loop spaces of their total manifolds are all homotopy equivalent.