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Non-smoothable homeomorphisms of $4$-manifolds with boundary

Daniel Galvin, Roberto Ladu

TL;DR

The paper proves that there exist non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary that fix the boundary and act trivially on homology, by constructing infinite families $(X_n,\partial X_n)$ and $(Z_n,\partial Z_n)$ with infinite Torelli groups and non-smoothable elements. It develops a Torelli-variation framework, including a gluing lemma, to detect non-smoothability and to relate boundary data to obstructions via Seiberg–Witten invariants. It also shows that some smoothable Torelli elements cannot be realized by generalised Dehn twists, and identifies conditions under which Torelli elements can or cannot be generated by Dehn twists, yielding examples where all Torelli elements are smoothable but remain collar-unrealizable. Altogether, the work reveals a rich structure for mapping classes of 4-manifolds with boundary, linking topological, gauge-theoretic, and contact–symplectic techniques to Torelli-type phenomena and Dehn-twist realizations.

Abstract

We construct the first examples of non-smoothable self-homeomorphisms of smooth $4$-manifolds with boundary that fix the boundary and act trivially on homology. As a corollary, we construct self-diffeomorphisms of $4$-manifolds with boundary that fix the boundary and act trivially on homology but cannot be isotoped to any self-diffeomorphism supported in a collar of the boundary and, in particular, are not isotopic to any generalised Dehn twist.

Non-smoothable homeomorphisms of $4$-manifolds with boundary

TL;DR

The paper proves that there exist non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary that fix the boundary and act trivially on homology, by constructing infinite families and with infinite Torelli groups and non-smoothable elements. It develops a Torelli-variation framework, including a gluing lemma, to detect non-smoothability and to relate boundary data to obstructions via Seiberg–Witten invariants. It also shows that some smoothable Torelli elements cannot be realized by generalised Dehn twists, and identifies conditions under which Torelli elements can or cannot be generated by Dehn twists, yielding examples where all Torelli elements are smoothable but remain collar-unrealizable. Altogether, the work reveals a rich structure for mapping classes of 4-manifolds with boundary, linking topological, gauge-theoretic, and contact–symplectic techniques to Torelli-type phenomena and Dehn-twist realizations.

Abstract

We construct the first examples of non-smoothable self-homeomorphisms of smooth -manifolds with boundary that fix the boundary and act trivially on homology. As a corollary, we construct self-diffeomorphisms of -manifolds with boundary that fix the boundary and act trivially on homology but cannot be isotoped to any self-diffeomorphism supported in a collar of the boundary and, in particular, are not isotopic to any generalised Dehn twist.
Paper Structure (16 sections, 17 theorems, 33 equations, 3 figures)

This paper contains 16 sections, 17 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Background
  4. Outline
  5. Acknowledgements
  6. Variations
  7. Definitions
  8. The Torelli group
  9. Applying variations to closed manifolds
  10. Sufficient conditions for non-smoothability
  11. Constructing examples
  12. Family from Legendrian surgery
  13. Family from knot surgery
  14. Generalised Dehn twists
  15. Absolute non-smoothability and generalised Dehn twists
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists an infinite family of pairwise non-diffeomorphic compact, oriented, smooth, simply-connected $4$-manifolds $\{(X_n,\partial X_n)\}_{n\in\mathbb{N}}$ with connected boundary and $\mathop{\mathrm{Tor}}\nolimits(X_n,\partial X_n)$ of infinite order such that, for each $n$, all non-trivial

Figures (3)

  • Figure 1: A Legendrian link diagram in standard form for $X_n$. The knot $K_{1,n}$ with the specified orientation has $2n+3$ crossings, $2n+2$ right cusps and rotation number $2n$. The knot $K_{2,n}$ has rotation number $0$.
  • Figure 2: The pictures show, in a surgery presentation for $\partial X_n$ (every link component is $0$-framed), the two surfaces $\Sigma_{1,n}$ (on the left) and $\Sigma_{2,n}$ (on the right) for the case $n=1$. They have genera $g(\Sigma_{1,n})=2n+3$ and $g(\Sigma_{2,n})=7$.
  • Figure 3: (a) Kirby diagram for the $4$-manifold $Z$, (b) Kirby diagram showing an embedding of $Z$ into $K_3\# 2\overline{\mathbb{CP}}^2$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: saeki_2006
  • Lemma 2.4: saeki_2006
  • Definition 2.5
  • Proposition 2.6: saeki_2006,orson_powell_2023
  • Theorem 2.7: orson_powell_2023
  • Remark 2.8
  • ...and 24 more