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

TL;DR

The paper investigates the loop-homotopy types of sphere bundles $S^{n}\to M\to N$ over simply connected $4$-manifolds $N$. It combines a cofiber/ Postnikov-style analysis of the base with obstruction-theoretic invariants $\omega_{2}$ and $p_{1}$ (and $e$ when $n=3$) to decompose the classifying map and then applies loop-space decomposition techniques (Hilton–Milnor, Bott–Samelson) to obtain explicit decompositions of $\Omega M$ depending on $d=\mathrm{rank}\,H^{2}(N;\mathbb{Z})$. The main results show that for $(d,n)\neq (0,2),(0,3)$, one has $\Omega M\simeq \Omega S^{n}\times \Omega N$, with refined products given for $d=0$, $d=1$, and $d\ge 2$. Consequently, $\pi_{*}(M)\cong \pi_{*}(S^{n})\oplus \pi_{*}(N)$ and $H_{*}(\Omega M)\cong H_{*}(\Omega S^{n})\otimes H_{*}(\Omega N)$, illustrating that infinitely many inequivalent sphere bundles yield loop-spaces that are homotopy equivalent. The paper also discusses exceptional cases, non-simply connected bases, and related questions on characteristic-class viewpoints.

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.

Theorems & Definitions (21)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof