Dax invariants, light bulbs, and isotopies of symplectic structures

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.

💡 Research Summary

The paper tackles several longstanding isotopy problems in four‑dimensional topology and symplectic geometry by introducing a new invariant, a generalization of the Dax invariant, for closed embedded surfaces. The authors first consider the product manifold M = Σ × S² where Σ is a closed oriented surface of positive genus. They study embeddings of Σ that are geometrically dual to the sphere G = {pt} × S². Building on Gabai’s 4‑dimensional light‑bulb theorem, which requires the “G‑inessential” condition to guarantee isotopy, the authors construct infinitely many embeddings that are homotopic (even relative to a neighborhood of G) but pairwise non‑isotopic, thereby providing a negative answer to Gabai’s question about removing the G‑inessential hypothesis.

The construction proceeds by fixing a standard embedding I₀ : Σ → Σ × {b₁} and inserting a 3‑ball D³ → M disjoint from both Σ₀ = I₀(Σ) and G. A sequence of embedded arcs γ_i, each intersecting D transversely once, is used to attach tubes to Σ₀, producing a family Σ_i = Σ₀ # γ_i. Each Σ_i shares the same geometric dual G and is homotopic to Σ₀, yet the authors prove that no smooth isotopy exists between distinct Σ_i. The key tool is the relative Dax invariant Dax(i₁,i₂) ∈ ℤ