Lightbulb theorem, embedded surfaces and isotopy of symplectic structures

YMSC Topology Seminar

Organizers：

陈伟彦、高鸿灏、黄意、林剑锋、邱宇、孙巍峰

Speaker:

Jianfeng Lin 林剑锋（YMSC）

Time：

Mon., 16:00-17:00, Sept. 15, 2025

Venue：

C654, Shuangqing Complex Building A

Online：

Zoom meeting ID: 405 416 0815

Passcode: 111111

Ttile:

Lightbulb theorem, embedded surfaces and isotopy of symplectic structures

Abstract:

Gabai's lightbulb theorem classifies embedded spheres in 4-manifolds with a geometric dual sphere. It is a breakthrough in 4-dimensional topology. In this talk, I discuss a joint work with Weiwei Wu, Yi Xie and Boyu Zhang, which classifies the isotopy classes of embeddings of a surface F into the product manifold F cross S2 with a geometric dual. This answers a question of Gabai regarding the generalized lightbulb theorem. Second, we show that the space of symplectic forms on an irrational ruled surface in a fixed cohomology class 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 that the smooth mapping class group of a surface cross S2 is infinitely generated.