Ring structures in singular instanton homology

TL;DR

This paper computes the ring structure of singular instanton Floer homology for certain 3-manifolds and proves an excision formula, resolving the last open case in the theory.

Contribution

It provides the first explicit calculation of the ring structure for these homologies and establishes an excision formula for the case n=1, completing the known cases.

Findings

Calculated the ring structure of singular instanton Floer homology for specific manifolds.

Proved an excision formula for singular instanton homology when n=1.

Resolved the last open case of excision in instanton Floer homology.

Abstract

We calculate the ring structure of the singular instanton Floer homology of $(S^1\times \Sigma, S^1\times \{p_1,\dots,p_n\})$ with C-coefficients, where $\Sigma$ is a closed oriented surface. As an application, we prove an excision formula for singular instanton homology when n=1. This settles the last unknown case of excision formula for instanton Floer homology.