Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

TL;DR

This paper develops an obstruction using instanton Floer cohomology to determine when homology 3-spheres can embed into homology S^3×S^1, employing covering spaces and instantons.

Contribution

It introduces a new cohomological obstruction based on instanton Floer theory for embeddings of homology 3-spheres into homology S^3×S^1.

Findings

Constructed an obstruction in filtered instanton Floer cohomology.

Applied the obstruction to specific embedding problems.

Utilized the $bZ$-fold covering space and instantons on it.

Abstract

We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold covering space of homology $S^3\times S^1$ and the instantons on it.