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A general connected sum formula for the families Seiberg-Witten invariant

Joshua Tomlin

TL;DR

The paper introduces a general connected sum formula for families Seiberg–Witten invariants by linking the families invariant to the Bauer–Furuta stable cohomotopy class from Part1 and reformulating the invariant in $S^1$-equivariant cohomology via Baraglia–Konno. A localised, cohomological approach is developed to compute invariants of smash products of monopole maps, enabling a fibrewise connected sum analysis. The main result expresses $SW^{f,\phi}_m$ for a fibrewise smash product as $SW^{f_2,\phi_2}(x^m\deg_{S^1}(f_1))$, and, together with a Fibrewise Bauer–Furuta formula, yields $[ ext{μ}]=[ ext{μ}_1]\wedge_{ ext{𝒥}}[ ext{μ}_2]$, giving a broad, chamber-compatible connected sum formula. This extends Baraglia–Konno’s gluing results, recovers the blow-up case when a fibre is $\overline{\mathbb{CP}}^2$, and provides a framework for detecting smooth isotopy phenomena via mapping tori in simply connected 4-manifolds.

Abstract

In ordinary Seiberg-Witten theory, there are well known connected sum formulae such as the vanishing formula and the blow up formula. For families Seiberg-Witten theory, there are results such as Liu's families blow-up formula and Baraglia-Konno's gluing formula, but these have limited uses. In this paper, we prove a general connected sum formula which incorporates these results. This is subsequent work to a previous paper [arXiv:2510.14201] in which we proved a connected sum formula for the Bauer-Furuta invariant.

A general connected sum formula for the families Seiberg-Witten invariant

TL;DR

The paper introduces a general connected sum formula for families Seiberg–Witten invariants by linking the families invariant to the Bauer–Furuta stable cohomotopy class from Part1 and reformulating the invariant in -equivariant cohomology via Baraglia–Konno. A localised, cohomological approach is developed to compute invariants of smash products of monopole maps, enabling a fibrewise connected sum analysis. The main result expresses for a fibrewise smash product as , and, together with a Fibrewise Bauer–Furuta formula, yields , giving a broad, chamber-compatible connected sum formula. This extends Baraglia–Konno’s gluing results, recovers the blow-up case when a fibre is , and provides a framework for detecting smooth isotopy phenomena via mapping tori in simply connected 4-manifolds.

Abstract

In ordinary Seiberg-Witten theory, there are well known connected sum formulae such as the vanishing formula and the blow up formula. For families Seiberg-Witten theory, there are results such as Liu's families blow-up formula and Baraglia-Konno's gluing formula, but these have limited uses. In this paper, we prove a general connected sum formula which incorporates these results. This is subsequent work to a previous paper [arXiv:2510.14201] in which we proved a connected sum formula for the Bauer-Furuta invariant.
Paper Structure (10 sections, 20 theorems, 93 equations)

This paper contains 10 sections, 20 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Families Seiberg-Witten invariants
  3. Generalised monopole maps
  4. Finite dimensional monopole maps
  5. Formulation in equivariant cohomology
  6. Equivariant cohomology
  7. Cohomology of sphere bundles
  8. Cohomological formula for $SW^{f,\phi}_m$
  9. Localised formula
  10. Seiberg-Witten connected sum formula

Key Result

Theorem 1.1

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

Theorems & Definitions (44)

  • Theorem 1.1: Part1 Theorem 6.8
  • Theorem 1.2: Families Seiberg-Witten Connected Sum Formula
  • Example 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8: BaragliaKonnoBFandSW Proposition 2.18
  • ...and 34 more