Dax invariants, light bulbs, and isotopies of symplectic structures

TL;DR

The paper advances isotopy theory on 4-manifolds by extending the Dax invariant to closed embedded surfaces in $M=\Sigma\times S^2$ and employing it to distinguish non-isotopic dual-sphere embeddings of $\Sigma$ in $M$. It develops the spinning-family and barbell-diffeomorphism machinery to generate independent elements in the mapping class group and to control embedding spaces via a fibration tower, culminating in a robust dual-surface classification via the Dax invariant. In the symplectic realm, the authors prove non-uniqueness phenomena on irrational ruled surfaces: the space of symplectic forms within a fixed cohomology class has infinitely many components, and there exist infinite formally-homotopic but not homotopic forms, with implications for $h$-principle questions in dimension four. Together, these results yield new insights into the structure of the mapping class group, the isotopy classification of dual-sphere embeddings, and the landscape of symplectic structures on closed 4-manifolds, highlighting the central role of the generalized Dax invariant.

Abstract

This paper addresses several isotopy problems on $4$-manifolds. First, we classify the isotopy classes of embeddings of $Σ$ in $Σ\times S^2$ that are geometrically dual to $\{\mbox{pt}\}\times S^2$, where $Σ$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic, thereby answering a question of Gabai. By combining this construction with techniques from symplectic topology, we also answer Problem 2(a) in McDuff-Salamon's problem list and a question of Cieliebak-Eliashberg-Mishachev, which concern the uniqueness and $h$-principle of symplectic structures on closed $4$-manifolds. We answer these questions by establishing the following results: (1) The space of symplectic forms on every irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components; (2) There exist infinitely many symplectic forms on every irrational ruled surface that are formally homotopic, cohomologous, but not homotopic to each other. Both are the first such examples for closed $4$-manifolds. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. In the course of the proof, we obtain several properties of the smooth mapping class group of $Σ\times S^2$, which may be of independent interest. For example, we show that there exists a surjective homomorphism from $π_0\operatorname{Diff}(Σ\times S^2)$ to $\mathbb{Z}^\infty$, such that its restriction to the subgroup of elements pseudo-isotopic to the identity is of infinite rank.

Figures (5)

• Figure 1: Example \ref{['example_self-referential']}
• Figure 2: Definitions \ref{['defi: general spining family']}
• Figure 3: The Definition of $\hat{\tau}_g$
• Figure 4: The construction of $m_k\# S_0$
• Figure 5: Defining data of a meridian-vertical barbell

Theorems & Definitions (166)

• Theorem 1.1
• Theorem : Gabai20
• Example 1.2
• Theorem 1.3
• Theorem 1.5
• Corollary 1.6
• proof
• Remark 1.7
• Theorem 1.8
• Definition 2.1