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The Dehn twist on a connected sum of two homology tori

Haochen Qiu

TL;DR

This work extends the Bauer–Furuta family invariant to nonsimply connected 4-manifolds via a quotient refinement of the $S^1$-equivariant invariant and analyzes the connected sum of two homology tori. By computing the invariants for the homology tori and their connected sum, the authors show the nonequivariant invariant vanishes under odd determinant hypotheses, while the quotient $S^1$-equivariant invariant remains nontrivial, detected by Hopf-type elements. The key steps include computing the index bundle, using gluing theorems, and applying an equivariant Hopf argument to produce a nontrivial element in the stable cohomotopy group, which implies the Dehn twist along a neck 3-sphere is not smoothly isotopic to the identity. The results generalize Kronheimer–Mrowka’s earlier work on simply connected manifolds to a broader nonsimply connected setting, providing new tools for detecting nontrivial smooth mapping class group elements in 4-manifolds. The refinement to a quotient invariant and the explicit computation in the two-torus case pave the way for further applications to nonsimply connected 4-manifolds and related mapping class problems.

Abstract

Kronheimer-Mrowka shows that the Dehn twist along a $3$-sphere in the neck of two $K3$ surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is $16 \mod 32$. We generalize the Pin$(2)$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if $X_1,X_2$ are two homology tori such that the determinants $r_1,r_2$ of them are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\# X_2$ is not smoothly isotopic to the identity.

The Dehn twist on a connected sum of two homology tori

TL;DR

This work extends the Bauer–Furuta family invariant to nonsimply connected 4-manifolds via a quotient refinement of the -equivariant invariant and analyzes the connected sum of two homology tori. By computing the invariants for the homology tori and their connected sum, the authors show the nonequivariant invariant vanishes under odd determinant hypotheses, while the quotient -equivariant invariant remains nontrivial, detected by Hopf-type elements. The key steps include computing the index bundle, using gluing theorems, and applying an equivariant Hopf argument to produce a nontrivial element in the stable cohomotopy group, which implies the Dehn twist along a neck 3-sphere is not smoothly isotopic to the identity. The results generalize Kronheimer–Mrowka’s earlier work on simply connected manifolds to a broader nonsimply connected setting, providing new tools for detecting nontrivial smooth mapping class group elements in 4-manifolds. The refinement to a quotient invariant and the explicit computation in the two-torus case pave the way for further applications to nonsimply connected 4-manifolds and related mapping class problems.

Abstract

Kronheimer-Mrowka shows that the Dehn twist along a -sphere in the neck of two surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is . We generalize the Pin-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if are two homology tori such that the determinants of them are odd, then the Dehn twist along a -sphere in the neck of is not smoothly isotopic to the identity.
Paper Structure (6 sections, 12 theorems, 68 equations)

This paper contains 6 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The family Bauer-Furata invariant for nonsimply connected manifolds
  3. The Bauer-Furuta invariant of homology tori
  4. Family Bauer-Furuta invariant of the connected sum of two tori
  5. Equivariant family Bauer-Furuta invariant of the connected sum of two tori
  6. Computation of the quotient $S^1$-equivariant Bauer-Furuta invariant

Key Result

Theorem 1.1

If $X1,X_2$ are two homology $4$-tori such that the determinants $r_1,r_2$ of them are odd. Then the Dehn twist along a $3$-sphere in the neck of $X_1\# X_2$ is not smoothly isotopic to the identity.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.7
  • Proposition 2.8: BF02 Corollary 3.2
  • ...and 13 more