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Exotic diffeomorphisms on a contractible 4-manifold surviving two stabilizations

Sungkyung Kang, JungHwan Park, Masaki Taniguchi

TL;DR

The paper develops a $\mathrm{Pin}(2) \times \mathbb{Z}_2$-equivariant refinement of lattice homotopy types to compute equivariant Seiberg--Witten Floer homotopy types and applies this framework to construct an exotic boundary Dehn twist on a Mazur-type contractible 4-manifold that persists after two stabilizations. By combining a connected-sum obstruction technique with a detailed equivariant lattice analysis and the Barraglia–Hekmati framework, the authors produce a new example where two stabilizations fail to trivialize the smooth phenomenon, and they compute explicit obstructions via a $\mathrm{Pin}(2) \times \mathbb{Z}_2$-equivariant spectrum. A suite of combinatorial tools, including $S^1\times\mathbb{Z}_p$-equivariant lattice types and large-prime Frøyshov-type invariants, underpins the obstruction theory and links topological stabilization to algebraic invariants. The results illuminate the sharp distinction between topological and smooth stabilization in dimension four and provide a concrete, computable mechanism for detecting exotic diffeomorphisms that survive two stabilizations.

Abstract

We develop a $\mathrm{Pin}(2) \times \mathbb{Z}_2$-equivariant refinement of the lattice homotopy type for computing equivariant Seiberg--Witten Floer homotopy types. As an application, we construct a relatively exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations.

Exotic diffeomorphisms on a contractible 4-manifold surviving two stabilizations

TL;DR

The paper develops a -equivariant refinement of lattice homotopy types to compute equivariant Seiberg--Witten Floer homotopy types and applies this framework to construct an exotic boundary Dehn twist on a Mazur-type contractible 4-manifold that persists after two stabilizations. By combining a connected-sum obstruction technique with a detailed equivariant lattice analysis and the Barraglia–Hekmati framework, the authors produce a new example where two stabilizations fail to trivialize the smooth phenomenon, and they compute explicit obstructions via a -equivariant spectrum. A suite of combinatorial tools, including -equivariant lattice types and large-prime Frøyshov-type invariants, underpins the obstruction theory and links topological stabilization to algebraic invariants. The results illuminate the sharp distinction between topological and smooth stabilization in dimension four and provide a concrete, computable mechanism for detecting exotic diffeomorphisms that survive two stabilizations.

Abstract

We develop a -equivariant refinement of the lattice homotopy type for computing equivariant Seiberg--Witten Floer homotopy types. As an application, we construct a relatively exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations.
Paper Structure (39 sections, 108 theorems, 881 equations, 2 figures)

This paper contains 39 sections, 108 theorems, 881 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background materials
  3. Notations
  4. Computation sequences and graded roots
  5. Graded roots from $\Delta$-sequences
  6. Montague's equivariant spectrum class
  7. Homotopy coherent Bauer--Furuta invariants
  8. Families categories
  9. Homotopy coherent bordism maps
  10. Equivariant Bauer--Furuta theory for equivariant $\mathrm{Spin}^c$ 4-manifolds
  11. $S^1\times \mathbb{Z}_p$-equivariant Seiberg--Witten Floer homotopy type
  12. Recovering Baraglia--Hekmati's theory
  13. Equivariant Bauer--Furuta invariants
  14. Equivariant lattice homotopy type
  15. $S^1$-action on plumbed 4-manifolds
  16. ...and 24 more sections

Key Result

Theorem 1.1

There exists a smooth compact contractible $4$-manifold $X$ with nonempty boundary, and an infinite family of relative diffeomorphisms $\{f_i \colon X \to X\}_{i \in \mathbb{N}}$ satisfying the following properties:

Figures (2)

  • Figure 1: The Mazur manifold bounded by $\Sigma(3,5,19)$.
  • Figure 2: Left: the graded root of $Y/\mathbb{Z}_2$ with respect to its canonical $\mathrm{Spin}^c$ structure. Right: the graded root of $Y/\mathbb{Z}_2$ with respect to the other $\mathrm{Spin}^c$ structure.

Theorems & Definitions (304)

  • Theorem 1.1
  • Theorem 1.2: \ref{['thm: eqv lattice comparison map']}
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 294 more