Surgery and Excision for Furuta-Ohta invariants on Homology $S^1 \times S^3$

TL;DR

This paper establishes surgery and excision formulas for the Furuta-Ohta invariant on homology S^1 x S^3, supporting its conjectured equivalence with the Seiberg-Witten invariant and computing it for specific mapping tori.

Contribution

It introduces new surgery and excision formulas for the Furuta-Ohta invariant and applies them to compute the invariant for certain 4-manifolds, advancing understanding of its properties.

Findings

Proved surgery and excision formulas for the Furuta-Ohta invariant.

Computed the invariant for specific mapping tori of 3-manifolds.

Provided a detailed description of ASD instanton moduli spaces on certain 4-manifolds.

Abstract

We prove a surgery formula and an excision formula for the Furuta-Ohta invariant $\lambda_{FO}$ defined on homology $S^1 \times S^3$, which provides more evidence on its equivalence with the Casson-Seiberg-Witten invariant $\lambda_{SW}$. These formulae are applied to compute $\lambda_{FO}$ of certain families of manifolds obtained as mapping tori under diffeomorphisms of $3$-manifolds. In the course of the proof, we give a complete description of the degree-zero moduli space of ASD instantons on $4$-manifolds of homology $H_*(D^2 \times T^2; \mathbb{Z})$ with a cylindrical end modeled on $[0, \infty) \times T^3$.