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Khovanov homology can distinguish exotic Mazur manifolds

Gheehyun Nahm

TL;DR

The paper introduces a skein-lasagna-free method to detect exotic 4-manifolds using Khovanov cobordism maps, building on Ren–Willis' framework. It applies these ideas to construct exotic Mazur manifolds by distinguishing exteriors of exotic disks, proving non-diffeomorphism while preserving homeomorphism, and providing explicit handle presentations. An alternative skein-lasagna route is also given, alongside an analysis-free argument that leverages Ren’s torus-link computations. An appendix uses SnapPy to verify boundary mapping class group triviality for related 3-manifolds, reinforcing the robustness of the approach. Overall, the work delivers the first analysis-free proof of exotic Mazur manifolds and broadens the utility of Khovanov theory in distinguishing smooth structures on 4-manifolds.

Abstract

In a recent breakthrough, Ren and Willis gave the first analysis-free proof of the existence of exotic compact, orientable 4-manifolds; their main tool is the Khovanov skein lasagna module defined by Morrison, Walker, and Wedrich. In this paper, we introduce a new, simple way of using Khovanov homology to distinguish certain exotic compact, orientable 4-manifolds; our new method does not depend on the skein lasagna module. As an application, we give the first analysis-free proof of the existence of exotic Mazur manifolds, i.e. compact, contractible 4-manifolds that have handle decompositions with a single 1-handle and a single 2-handle.

Khovanov homology can distinguish exotic Mazur manifolds

TL;DR

The paper introduces a skein-lasagna-free method to detect exotic 4-manifolds using Khovanov cobordism maps, building on Ren–Willis' framework. It applies these ideas to construct exotic Mazur manifolds by distinguishing exteriors of exotic disks, proving non-diffeomorphism while preserving homeomorphism, and providing explicit handle presentations. An alternative skein-lasagna route is also given, alongside an analysis-free argument that leverages Ren’s torus-link computations. An appendix uses SnapPy to verify boundary mapping class group triviality for related 3-manifolds, reinforcing the robustness of the approach. Overall, the work delivers the first analysis-free proof of exotic Mazur manifolds and broadens the utility of Khovanov theory in distinguishing smooth structures on 4-manifolds.

Abstract

In a recent breakthrough, Ren and Willis gave the first analysis-free proof of the existence of exotic compact, orientable 4-manifolds; their main tool is the Khovanov skein lasagna module defined by Morrison, Walker, and Wedrich. In this paper, we introduce a new, simple way of using Khovanov homology to distinguish certain exotic compact, orientable 4-manifolds; our new method does not depend on the skein lasagna module. As an application, we give the first analysis-free proof of the existence of exotic Mazur manifolds, i.e. compact, contractible 4-manifolds that have handle decompositions with a single 1-handle and a single 2-handle.

Paper Structure

This paper contains 7 sections, 5 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Exotic compact, orientable $4$-manifolds from exotic disks
  3. Khovanov homology and oriented surfaces in CP2\\int(D4)
  4. A direct argument
  5. A skein lasagna argument
  6. Exotic Mazur manifolds from Khovanov homology
  7. Computation of the mapping class group

Key Result

Theorem 1.1

For every integer $k\ge1$, Khovanov homology can distinguish the exotic pair of Mazur manifolds of Figure fig:mazur.

Figures (5)

  • Figure 1.1: Exotic pairs of Mazur manifolds ($k\in\mathbb{Z}$, $k\ge1$).
  • Figure 2.1: MR4726569 The exotic slice disks $\Sigma_{k},\Sigma_{k}'$ of $J_{k}$
  • Figure 2.2: Handle diagrams of the exotic pairs of disk exteriors $D^{4}\backslash N(\Sigma_{k})$ and $D^{4}\backslash N(\Sigma_{k}')$
  • Figure 4.1: Handle calculus that shows that the two manifolds of Figure \ref{['fig:mazur']} are diffeomorphic to the disk exteriors $-((\mathbb{CP}^{2})^{\circ}\backslash N(S_{k}))$ and $-((\mathbb{CP}^{2})^{\circ}\backslash N(S_{k}'))$ from the proof of Theorem \ref{['thm:Khovanov-homology-can']}. In the final step, the handle diagrams for $k=1$ are shown.
  • Figure A.1: MR4726569 A $2$-component link $J_{0}\sqcup U$ancillaryfiles.

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.4
  • Theorem 2.1: MR4726569 and proof of MR4726569
  • Remark 2.2
  • Lemma 2.3: MWW, MR4504654, 334386
  • proof
  • Corollary 2.4
  • proof
  • Remark 3.1
  • Remark 3.2
  • ...and 5 more