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

TL;DR

This paper classifies the total spaces of $S^3$-bundles over $ ext{CP}^2$ up to orientation-preserving homotopy equivalence by computing invariants and applying surgery theory.

Contribution

It provides a complete classification of these bundles' total spaces up to homotopy, combining Kreck-Stolz invariants with surgery theory.

Findings

PL-homeomorphism classification via Kreck-Stolz invariants

Homotopy classification using surgery theory

Explicit classification results for $S^3$-bundles over $ ext{CP}^2"

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.