Surgery and excision for Furuta-Ohta invariants on homology ${S}^1 \times {S}^3$

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 establish the existence of asymptotic values for finite energy instantons over end-cylindrical manifolds perturbed by exponentially decay holonomy perturbations, and give a complete description of the structure of the moduli space of charge-zero instantons over any non-compact $4$-manifold whose end is modeled on $[0,\infty) \times T^3$, and whose homology is given by $H_\ast (D^2 \times T^2; \mathbb{Z})$.