📝 SummarXiv

Abstract

This paper proves the following two main results. 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. This answers a question of Gabai. Second, we show that the space of symplectic forms on an irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components. This gives the first such example among closed $4$--manifolds and answers Problem 2(a) in McDuff--Salamon's problem list. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. In the proof, we also establish 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 $\operatorname{MCG}(\Sigma\times S^2)$ to $\mathbb{Z}^\infty$, such that its restriction to the subgroup of elements pseudo-isotopic to the identity is also of infinite rank.