Differential Geometry, Lie Theory and Low-Dimensional Topology, Melbourne, Australia, 2016

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Details of talk

TitleIndefinite fibrations on 4-manifolds
PresenterOsamu Saeki (Kyushu University)
Author(s)Prof Osamu Saeki

A broken Lefschetz fibration (BLF, for short) is a smooth map of a closed
oriented 4-manifold onto a closed surface whose singularities consist of
Lefschetz critical points together with indefinite folds (or round
singularities). Such a class of maps was first introduced by
Auroux-Donaldson-Katzarkov in relation to near-symplectic structures. In this
talk, we give a set of explicit moves for indefinite fibrations, which include
BLFs and indefinite generic maps, and give an elementary and constructive proof
to the fact that any map into the 2-sphere is homotopic to a BLF with embedded
round image. We also show how to realize any given null-homologous 1-dimensional
submanifold with prescribed local models for its components as the round locus
of a BLF. These algorithms allow us to give a purely topological and
constructive proof of a theorem of Auroux-Donaldson-Katzarkov on the existence
of broken Lefschetz pencils with embedded round image on near-symplectic
4-manifolds. This is a joint work with R. \.{I}nan\c{c} Baykur (University of