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

Langte Ma

Abstract

We prove a surgery formula and an excision formula for the Furuta-Ohta invariant $λ_{FO}$ defined on homology $S^1 \times S^3$, which provides more evidence on its equivalence with the Casson-Seiberg-Witten invariant $λ_{SW}$. These formulae are applied to compute $λ_{FO}$ 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 $[0, \infty) \times T^3$.

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 defined on homology , which provides more evidence on its equivalence with the Casson-Seiberg-Witten invariant . These formulae are applied to compute of certain families of manifolds obtained as mapping tori under diffeomorphisms of -manifolds. In the course of the proof, we give a complete description of the degree-zero moduli space of ASD instantons on -manifolds of homology with a cylindrical end modeled on .

Paper Structure

This paper contains 28 sections, 50 theorems, 342 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Unperturbed Moduli Spaces
  3. Moduli Spaces of Instantons over $4$-Manifolds
  4. Moduli Space of Flat connections over $3$-Manifolds
  5. Holonomy Perturbations
  6. Perturbations over a Compact Region
  7. Perturbations over Cylindrical Ends
  8. Gauge Fixing and Regularity
  9. The Morgan--Mrowka--Ruberman Gauge Fixing
  10. Regularity Estimates
  11. The Asymptotic Behaviors
  12. The Asymptotic Map
  13. A Refinement of the Asymptotic Map
  14. Compactness of $\mathcal{M}_{\sigma}(Z)$
  15. Transversality on the Irreducible Moduli Space
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $X$ be an admissible homology $S^1 \times S^3$ and $\mathcal{T} \subset X$ an essentially embedded torus. Given an integer $q \in \mathbb{Z}$, after fixing a generator $1_X \in H^1(X; \mathbb{Z})$ the Furuta--Ohta invariant of the $(1, q)$-surgered manifold $X_{1, q}$ is related to the Furuta--O

Figures (1)

  • Figure 1: The Cube Portion $\mathcal{C}^o_{T^3}$

Theorems & Definitions (114)

  • Definition 1
  • Theorem 1.1
  • Corollary 1
  • Proposition 1
  • Theorem 1.2
  • Corollary 2
  • Theorem 1.3
  • Remark 1
  • Remark 2
  • Theorem 1.4
  • ...and 104 more