Pullbacks of Sphere Fibrations over Connected Sums

TL;DR

This paper develops new homotopy-theoretic methods to analyze when pullbacks of sphere fibrations over connected sums are homotopy equivalent to connected sums with gyrations, extending previous geometric approaches.

Contribution

It introduces purely homotopy-theoretic techniques to generalize results from manifolds to Poincaré complexes and from integral to local settings.

Findings

Established conditions for homotopy equivalence of pullback total spaces

Extended results to Poincaré Duality complexes

Applied methods to various topological scenarios

Abstract

We prove conditions under which the total space of the pullback of a sphere fibration over a connected sum is homotopy equivalent to a connected sum with a gyration. Existing results of this type often depend on geometric methods. We develop new methods based only on homotopy theory, allowing for generalisations from manifolds to Poincar\'e Duality complexes and from integral settings to local ones. Several applications are given.