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<p>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.</p>

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Sphere bundles over 4-manifolds are trivial after looping

  • 期刊论文

  • 发表日期：

    2025年5月7日

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  • Ruizhi Huang

摘要

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.