A user's guide to building ribbon surfaces and holomorphic curves in 4-manifolds

A user's guide to building ribbon surfaces and holomorphic curves in 4-manifolds

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Kyle Hayden
Taplin Auditorium

We'll review a variety of hands-on ways to build ribbon surfaces in 4-manifolds, with an eye towards building holomorphic curves and symplectic/Lagrangian surfaces in Stein domains. Inspired by a simple construction of interesting ribbon disks, we build unknotted holomorphic and Lagrangian disks in the 4-ball with

(surprisingly!) nontrivial cross-sections. Then we'll introduce another simple construction of ribbon surfaces in 4-manifolds using braids and open book decompositions of 3-manifolds. As an application, we show that every quasipositive braid in a positive open book for a 3-manifold Y bounds a holomorphic curve in some Stein filling of Y. (The latter is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)