Stable classification of 4-manifolds obtained by the surgery on loops

TL;DR

This paper investigates the stable diffeomorphism classification of 4-manifolds created via loop surgery, showing that under certain conditions, the signature is the sole invariant needed for classification.

Contribution

It establishes that the signature completely classifies certain 4-manifolds obtained by loop surgery when their normal 1-types coincide.

Findings

Signature is the only invariant under the given conditions.

Certain 4-manifolds with the same fundamental group are stably diffeomorphic.

Classification simplifies to signature comparison in these cases.

Abstract

This paper is concerned with the problem of stable diffeomorphism classification of 4-manifolds obtained using the surgery on loops. The main theorem states that under the assumption that the normal 1-type of two 4-manifolds in question is the same, the only classifying invariant is the signature. In particular, in some cases, any two closed smooth 4-manifolds with a given fundamental group, obtained by the standard construction, are stably diffeomorphic.