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Surfaces in 4-manifolds and extendible mapping classes

Shital Lawande, Kuldeep Saha

TL;DR

This work investigates when mapping classes of embedded surfaces in 4-manifolds extend to ambient diffeomorphisms, introducing the notions of extendible and flexible embeddings and connecting them to open book embeddings and Rokhlin quadratic forms. The authors prove strong nonexistence results: in manifolds with the homology type of a 4-ball or 4-sphere, most $(g,b)$ cannot admit flexible embeddings, with only $(1,1)$ and $(0,2)$ as exceptions for embeddings into $\mathbb{D}^4$, and they show there is no universal simple open book of $\mathbb{S}^5$ with a spin page. They develop extensive criteria and constructions for extendible and non-extendible Dehn twists, including Seifert-type embeddings, Hopf annulus tricks, and fibered Dehn twists, and connect these to sliceness obstructions for links in homology 4-balls. A key application is that certain open books on $\mathbb{S}^5$ cannot be universal, while the fibered twist framework provides a generating set of extendible mapping classes with only $3g$ generators for the trivial genus $g$ embedding in $\mathbb{S}^4$, improving earlier bounds. Overall, the paper links embedding geometry, mapping class extendibility, open books, and 4-manifold topology, offering new tools and obstructions in high-dimensional contact and 4-manifold theory.

Abstract

We study smooth proper embeddings of compact orientable surfaces in compact orientable $4$-manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient $4$-manifolds. We call such mapping classes extendible. An embedding for which all mapping classes are extendible is called flexible. We show that for most of the surfaces there exists no flexible embedding in a $4$-manifold with homology type of a $4$-ball or of a $4$-sphere. As an application of our method, we address a question of Etnyre and Lekili and show that there exists no simple open book decomposition of $S^5$ with a spin page where all $3$-dimensional open books admit open book embeddings. We also provide many constructions and criteria for extendible and non-extendible mapping classes, and discuss a connection between extendibility and sliceness of links in a homology $4$-ball with $S^3$ boundary. Finally, we give a new generating set of the group of extendible mapping classes for the trivial embedding of a closed genus $g$ surface in $S^4$, consisting of $3g$ generators. This improves a previous result of Hirose giving a generating set of size $6g-1$.

Surfaces in 4-manifolds and extendible mapping classes

TL;DR

This work investigates when mapping classes of embedded surfaces in 4-manifolds extend to ambient diffeomorphisms, introducing the notions of extendible and flexible embeddings and connecting them to open book embeddings and Rokhlin quadratic forms. The authors prove strong nonexistence results: in manifolds with the homology type of a 4-ball or 4-sphere, most cannot admit flexible embeddings, with only and as exceptions for embeddings into , and they show there is no universal simple open book of with a spin page. They develop extensive criteria and constructions for extendible and non-extendible Dehn twists, including Seifert-type embeddings, Hopf annulus tricks, and fibered Dehn twists, and connect these to sliceness obstructions for links in homology 4-balls. A key application is that certain open books on cannot be universal, while the fibered twist framework provides a generating set of extendible mapping classes with only generators for the trivial genus embedding in , improving earlier bounds. Overall, the paper links embedding geometry, mapping class extendibility, open books, and 4-manifold topology, offering new tools and obstructions in high-dimensional contact and 4-manifold theory.

Abstract

We study smooth proper embeddings of compact orientable surfaces in compact orientable -manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient -manifolds. We call such mapping classes extendible. An embedding for which all mapping classes are extendible is called flexible. We show that for most of the surfaces there exists no flexible embedding in a -manifold with homology type of a -ball or of a -sphere. As an application of our method, we address a question of Etnyre and Lekili and show that there exists no simple open book decomposition of with a spin page where all -dimensional open books admit open book embeddings. We also provide many constructions and criteria for extendible and non-extendible mapping classes, and discuss a connection between extendibility and sliceness of links in a homology -ball with boundary. Finally, we give a new generating set of the group of extendible mapping classes for the trivial embedding of a closed genus surface in , consisting of generators. This improves a previous result of Hirose giving a generating set of size .

Paper Structure

This paper contains 16 sections, 17 theorems, 7 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Open books and embeddings
  4. Extendible mapping classes of embedded surfaces
  5. Characteristic embedding and Rokhlin quadratic form
  6. Rokhlin quadratic form for some surfaces in $\mathbb{D}^4$
  7. Generalized Dehn twists along $3$-manifolds
  8. Existence problem of flexible codimension 2 embedding of surface
  9. Proof of Theorem \ref{['thm1.5']}
  10. Examples of extendible/non-extendible mapping classes
  11. Mapping classes induced by an isotopy of $\mathbb{S}^3 \times [0,1]$
  12. Embeddings of $\Sigma_{g,1}$ and Hammenstädt generators
  13. Flexible embeddings and sliceness of links
  14. Extendible mapping classes induced via Fibered Dehn twists on $\mathbb{S}^4$
  15. A circle action on $\mathbb{S}^3$ via open book and $\textrm{T}_{\mathbb{S}^3}$
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $W^4$ be an orientable compact $4$-manifold with boundary (possibly empty). Assume that either $W$ is an integral homology $4$-ball with boundary an integral homology $3$-sphere, or $W$ is an integral homology $4$-sphere. For $g \geq 0$ and $b \geq 0$, let $\Sigma_{g,b}$ denote a compact orienta

Figures (12)

  • Figure 1: Standardly embedded $\Sigma_{7,1}$ inside $\mathbb{D}^4$
  • Figure 6: Hopf annuli
  • Figure 7: Embeddings of $\Sigma_{1,1}$ in $\mathbb{D}^4$ with boubdaries Trefoil knot and Fig-eight knot
  • Figure 9:
  • Figure 10: Hammenst$\ddot{a}$dt curves in $\Sigma_7$ for an odd spin structure.
  • ...and 7 more figures

Theorems & Definitions (40)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 2.1
  • Definition 2.1: Simple open book
  • Definition 2.2: Open book embedding
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 30 more