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

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.

14 pages; accepted version in Mathematische Zeitschrift