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A general connected sum formula for the families Bauer-Furuta invariant

Joshua Tomlin

Abstract

The Bauer-Furuta invariant of a family of smooth 4-manifolds is a stable cohomotopy refinement of the families Seiberg-Witten invariant and is constructed from a finite dimensional approximation of the Seiberg-Witten monopole map. We prove a general formula for the families Bauer-Furuta invariant of a fibrewise connected sum, extending Bauer's non-parameterised formula. In a subsequent paper, we will use this formula to derive a general connected sum formula for the families Seiberg-Witten invariant which incorporates both the families blow-up formula of Liu and the gluing formula of Baraglia-Konno.

A general connected sum formula for the families Bauer-Furuta invariant

Abstract

The Bauer-Furuta invariant of a family of smooth 4-manifolds is a stable cohomotopy refinement of the families Seiberg-Witten invariant and is constructed from a finite dimensional approximation of the Seiberg-Witten monopole map. We prove a general formula for the families Bauer-Furuta invariant of a fibrewise connected sum, extending Bauer's non-parameterised formula. In a subsequent paper, we will use this formula to derive a general connected sum formula for the families Seiberg-Witten invariant which incorporates both the families blow-up formula of Liu and the gluing formula of Baraglia-Konno.
Paper Structure (19 sections, 39 theorems, 263 equations)

This paper contains 19 sections, 39 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Finite dimensional approximation
  3. Bauer-Furuta Approximation
  4. Schwarz approximation
  5. Equivalence
  6. The families Bauer-Furuta invariant
  7. Families with separating necks
  8. The families Seiberg-Witten monopole map
  9. Monopoles on the neck
  10. Sobolev estimates
  11. Elliptic inequality
  12. Elliptic bootstrapping
  13. Exponential decay
  14. Proof of the Families Permutation Theorem
  15. The first homotopy
  16. ...and 4 more sections

Key Result

Theorem 1.1

For $j \in \{1,2\}$, let $E_j \to B$ be a smooth family of closed, oriented 4-manifolds equipped with a $\text{spin}^c\text{ }$ structure $\mathfrak{s}_j$ on the vertical tangent bundle. Assume a section $i_j : B \to E_j$ exists with normal bundle $V_j$ and suppose that $\varphi : V_1 \to V_2$ is an Then the families Bauer-Furuta class of the fiberwise connected sum $E = E_1 \#_B E_2$ is

Theorems & Definitions (96)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4: Suspension Spectrum
  • Example 2.5: Desuspension
  • Example 2.6: Smash product of spectra
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 86 more