Mathematics Colloquia and Seminars

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A Lefschetz fibration of $4$--manifold is a map $f: X \rightarrow D^2$ with finitely many singularities locally modeled by $(z,w) \mapsto z^2 +w^2$ and whose regular fibers are diffeomorphic surfaces with boundary. If $\partial X \neq \empty,$ then a Lefschetz fibration induces an open decomposition of $\partial X.$ I will discuss the topology of Lefschetz fibrations and how they relate to both open book decompositions of $3$--manifolds and relative trisections of $4$--manifolds with boundary.