On the Classification of $S^3$-Bundles over $\mathbb{C}P^2$

TL;DR

The paper resolves the orientation-preserving homotopy classification of the total spaces of $S^3$-bundles over $\mathbb{C}P^2$ with nonzero Euler class by combining a PL–homeomorphism classification via Kreck–Stolz invariants with a surgery-theoretic homotopy analysis. It shows that, for $l\neq 0$, homotopy types are determined by $k\equiv k'\pmod{6}$ (and similarly for the spin family), and it proves vanishing of $\pi_4$ for $M'_{k,l}$ when $l$ is odd, addressing conjectures in prior work. The approach systematically ties together vector-bundle data over $\mathbb{C}P^2$, explicit invariant calculations, and the PL surgery framework to obtain a complete homotopy classification, extending and clarifying earlier results on diffeomorphism and homeomorphism classifications. The results provide a coherent map from PL structures to homotopy types for these seven-manifolds and relate the algebraic invariants to geometric bundle data.

Abstract

This paper presents a classification of the total spaces of $S^3$-bundles over $\mathbb{C}P^2$ up to orientation-preserving homotopy equivalence. Our approach proceeds in two steps: we first derive the PL-homeomorphism classification for these manifolds by computing their Kreck-Stolz invariants. Then, building upon this PL classification result and through an application of surgery theory, we establish the homotopy equivalence classification.

Figures (4)

• Figure 1: Diagram 1
• Figure 2: Diagram 2
• Figure 3: Diagram 3
• Figure :

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4: Theorem 3.1 of wxq
• Theorem 2.1: Theorem 1 of kreck1998correction
• Theorem 2.2
• proof : Proof of Theorem \ref{['result']}
• proof : Proof of Corollary \ref{['cor:hmtpgp']}
• Lemma 3.1
• proof