arXiv subjects

Kenta Hayano

Publications and source records attributed to Kenta Hayano.

20 records · Page 2Linked to original sources

arXiv2011

A note on sections of broken Lefschetz fibrations

We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above conditions. Smith proved that every Lefschetz fibration has only finitely many homotopy classes of sections, and Smith and Stipsicz independently proved that a Lefschetz fibration is trivial if it has a section with non-negative square. So our results indicate that there are no generalizations of the above results to broken Lefschetz fibrations. We also give a necessary and sufficient condition for the total space of a simplified broken Lefschetz fibration with a section admitting a spin structure, which is a generalization of Stipsicz's result on Lefschetz fibrations.

math.GT↗

arXiv2011

On genus-1 simplified broken Lefschetz fibrations

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas' classification theorem of genus-1 Lefschetz fibrations, which states that the total space of a non-trivial genus-1 Lefschetz fibration over $S^2$ is diffeomorphic to an elliptic surface E(n), for some $n\geq 1$.

math.GT↗