Kenta Hayano

On genus-1 simplified broken Lefschetz fibrations

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefschetz 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 nontrivial genus‐1 Lefschetz fibration over S 2 is diffeomorphic to an elliptic surface E.n/ for some n 1. 57M50; 32S50, 57R65

Multisections of Lefschetz fibrations and topology of symplectic 4–manifolds

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the 4-ball. Various problems regarding the topology of symplectic 4-manifolds, such as the smooth classification of symplectic Calabi-Yau 4-manifolds, can be translated to combinatorial problems in this manner. After producing special monodromy factorizations of Lefschetz pencils on symplectic Calabi-Yau K3 and Enriques surfaces, and introducing monodromy substitutions tailored for generating multisections, we obtain several novel applications, allowing us to construct: new counter-examples to Stipsiczs conjecture on fiber sum indecomposable Lefschetz fibrations, non-isomorphic Lefschetz pencils of the same genera on the same new symplectic 4-manifolds, the very first examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.

Elimination of cusps in dimension 4 and its applications

We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M×S1, where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.

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 Stipsiczs result on Lefschetz fibrations.

Region crossing change on spatial-graph diagrams

Region crossing change at a region of a knot, link or spatial-graph diagram is a local transformation which changes all the crossings on the boundary of the region. In this paper, we show that we can make any crossing change by a finite number of region crossing changes on any diagram of a connected spatial graph which has no cutting circles.

Modification rule of monodromies in an R2-move

An R2 ‐move is a homotopy of wrinkled fibrations which deforms images of indefinite fold singularities like the Reidemeister move of type II. Variants of this move are contained in several important deformations of wrinkled fibrations. In this paper, we first investigate how monodromies are changed by this move. For a given fibration and its vanishing cycles, we then give an algorithm to obtain vanishing cycles in a single reference fiber of a fibration obtained by flip and slip, which is a sequence of homotopies increasing fiber genera. As an application of this algorithm, we give several examples of diagrams which were introduced by Williams to describe smooth 4‐manifolds by a finite sequence of simple closed curves in a closed surface. 57R45; 30F99