Surgery on tori in the 4-sphere

TL;DR

This paper studies how torus surgeries in the 4-sphere can produce various 4-manifolds, explores their uniqueness, and constructs new embeddings of 3-manifolds, including homology spheres, into 4-manifolds.

Contribution

It introduces methods for torus surgery in $S^4$, analyzes the resulting 4-manifolds, and provides new embeddings of 3-manifolds, expanding understanding of 4-dimensional topology.

Findings

Characterization of 4-manifolds obtainable via torus surgery

Results on the uniqueness of torus surgery descriptions

New embeddings of homology spheres into $S^4$

Abstract

We investigate the operation of torus surgery on tori embedded in $S^4$. Key questions include which 4-manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3-manifolds into 4-manifolds by viewing Dehn surgery as a cross section of a surgery on a surface. In particular, we give new embeddings of homology spheres into $S^4$.