4-manifolds, surgery on loops and geometric realization of Tietze transformations

TL;DR

This paper establishes that certain smooth closed 4-manifolds obtained by surgery on loops are stably diffeomorphic if and only if their signatures match, extending previous results on 4-manifold classification.

Contribution

It introduces a new criterion based on signatures for stable diffeomorphism of 4-manifolds obtained via loop surgery, using modified Kreck surgery and Kirby calculus.

Findings

Stable diffeomorphism characterized by signature equality.

Extension of Wall's classification results to a broader class of 4-manifolds.

Application of modified surgery and Kirby calculus techniques.

Abstract

In the paper \cite{wall_1}, C.T.C. Wall proved that two smooth closed simply connected 4-manifolds which are homeomorphic are in fact stably diffeomorphic. We prove a similar result which states that two smooth closed 4-manifolds satisfying certain properties are stably diffeomorphic if and only if their signatures agree. The manifolds in question are obtained by surgery on loops. The methods we use are modified surgery of Kreck \cite{kreck} and Kirby calculus.