s-Cobordism classification of $4$-manifolds through the group of homotopy self-equivalences

TL;DR

This paper classifies certain topological 4-manifolds up to s-cobordism using homotopy self-equivalence groups and modified surgery theory, extending understanding of 4-manifold invariants.

Contribution

It introduces a new classification method for 4-manifolds with fundamental group of cohomological dimension at most 2, utilizing braid constructions and Kreck's modified surgery.

Findings

Provides s-cobordism classification for specific 4-manifolds

Utilizes braid constructions of homotopy self-equivalences

Applies modified surgery theory to topological 4-manifolds

Abstract

The aim of this paper is to give an $s$-cobordism classification of topological $4$-manifolds in terms of the standard invariants using the group of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study the group of homotopy self-equivalences of $4$-manifolds. Using this braid together with the modified surgery theory of Kreck, we give an $s$-cobordism classification for certain $4$-manifolds with fundamental group $\pi$, such that cd $\pi \leq 2$.