Pochette surgery of 4-sphere

TL;DR

This paper investigates pochette surgery on 4-manifolds, particularly the 4-sphere, computing homology changes and identifying conditions under which the surgery preserves the manifold's diffeomorphism type.

Contribution

It provides homology computations for pochette surgeries on homology 4-spheres and characterizes when such surgeries preserve the 4-sphere, including cases with non-trivial cores and cords.

Findings

Pochette surgery with trivial cord does not change the 4-sphere.

Certain pochette surgeries yield the 4-sphere despite non-trivial cores and cords.

Homology of pochette surgeries can be computed using linking numbers.

Abstract

Iwase and Matsumoto defined `pochette surgery' as a cut-and-paste on 4-manifolds along a 4-manifold homotopy equivalent to $S^2\vee S^1$. The first author in [10] studied infinitely many homotopy 4-spheres obtained by pochette surgery. In this paper we compute the homology of pochette surgery of any homology 4-sphere by using `linking number' of a pochette embedding. We prove that pochette surgery with the trivial cord does not change the diffeomorphism type or gives a Gluck surgery. We also show that there exist pochette surgeries on the 4-sphere with a non-trivial core sphere and a non-trivial cord such that the surgeries give the 4-sphere.