University of Arkansas Topology Seminar: 10/24/2019

Speaker: Nickolas Castro

Title: Relative L–invariant for compact 4–manifolds II

In this talk, we will discuss a new integer valued invariant of compact, smooth, oriented 4–manifold with boundary X. The relative L–invariant is roughly defined by the minimal length of certain desirable paths in a subset of the arc and curve complex of a compact surface with boundary. This subcomplex is tailor made so that relative trisections of X correspond to paths in this complex which also retain the data of the induced open book decomposition of the boundary of X. We will show that there exist manifolds W such that rL(W) > m for any m. The main goal of the talk is to prove the following theorem: If B is a rational homology 4–ball and rL(B) = 0, then B is diffeomorphic to the standard 4–ball. This talk will include preliminary discussions on relative trisections of 4–manifolds and how to determine the induced open book decomposition on the bounding 3–manifold.



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