Infinite connected components of the space of symplectic forms on ruled surfaces

TL;DR

This paper constructs an infinite family of distinct symplectic forms on ruled surfaces, demonstrating that many non-isotopic symplectic structures can exist on the same four-manifold.

Contribution

It introduces an infinite set of pairwise non-isotopic symplectic forms on ruled surfaces, addressing a key question about the uniqueness of symplectic structures.

Findings

Existence of infinitely many non-isotopic symplectic forms on the same ruled surface

Counterexamples to symplectic uniqueness up to isotopy on closed four-manifolds

Advancement in understanding symplectic topology of ruled surfaces

Abstract

We provide an infinite family of diffeomorphic symplectic forms on ruled surfaces, which are pairwise non-isotopic. This answers a uniqueness question regarding symplectic structures up to isotopy on closed symplectic four-manifolds.