USC Topology Seminar: Decompositions and diagrams of symplectic surfaces in Weinstein domains

Melissa Zhang, UC Davis

Title: Decompositions and diagrams of symplectic surfaces in Weinstein domains

Abstract: A closed 3-manifold can be diagrammatically encoded by a Heegaard diagram, which comes from a decomposition of the space into 1-handlebodies called a Heegaard splitting. Analogously, Gay and Kirby's trisections diagrammatically encode the decomposition of a smooth 4-manifold into three 1-handlebodies. More generally, Islambouli--Naylor multisections decompose 4-manifolds with boundary into n 1-handlebodies. Meier--Zupan's bridge trisections encode surfaces in these 4-manifolds.

We may ask for decompositions that further encode geometric structure. In 2023, Islambouli and Starkston defined "multisections with divides", which additionally capture the symplectic structure of a Weinstein domain. In joint work with Román Aranda, Patricia Cahn, and Agniva Roy, we introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli--Starkston's bisection-with-divides structure on the Weinstein domain. In this talk, I will introduce the decompositions and diagrams described above, and then show how one can algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations.