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 $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to $\{\mbox{pt}\}\times S^2$, where $\Sigma$ 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 $\Sigma\times S^2$, which may be of independent interest. For example, we show that there exists a surjective homomorphism from $\pi_0\operatorname{Diff}(\Sigma\times S^2)$ to $\mathbb{Z}^\infty$, such that its restriction to the subgroup of elements pseudo-isotopic to the identity is of infinite rank.