Sphere bundles over $4$-manifolds are trivial after looping
Algebraic Topology
2025-04-30 v2 Geometric Topology
Abstract
We show that except two special cases, the sphere bundle of a vector bundle over a simply connected -manifold splits after looping. In particular, this implies that though there are infinitely many inequivalent sphere bundles of a given rank over a -manifold, the loop spaces of their total manifolds are all homotopy equivalent.
Cite
@article{arxiv.2210.17352,
title = {Sphere bundles over $4$-manifolds are trivial after looping},
author = {Ruizhi Huang},
journal= {arXiv preprint arXiv:2210.17352},
year = {2025}
}
Comments
14 pages; accepted version in Mathematische Zeitschrift