Broad Audience Title

Constructing Interesting 4-Dimensional Spaces

Scientific Title

Constructing Lefschetz Fibrations on Symplectic 4-Manifolds via Postive Factorizations in the Mapping Class Group

By Kai Nakamura
Renewable Energy
iCons Year 4
2018
Executive Summary 

Symplectic manifolds originally arose in classical mechanics as the phase space of a closed system. However, they were not originally recognised as such and the modern formulation of a symplectic manifold was not realized until the 20th century. Since then symplectic manifolds have gain attention purely for their interesting mathematical properties. In the 1990’s, the study of symplectic 4-manifold gained a topological flavor with Simon Donaldson work showing that any symplectic 4-manifold has a Lefschetz pencil. Symplectic 4-manifolds and their Lefschetz pencils have provided a particularly rich category in low dimensional topology.

A Lefschetz pencil on a symplectic 4-manifold can be blown up to be turned into what is called a Lefschetz fibration. This a smooth map from a symplectic 4-manifold to a sphere which is a submersion except at finitely many critical points where the map has a particularly simple local model. At the smooth points, the fiber of this map will be a genus g surface and we call such a map a genus g Lefschetz fibration. It turns out we get more information about the symplectic 4-manifold by working around one of the critical points. The monodromy of the generic fiber around one of these critical points gives what is called a Dehn twist which is an element of the mapping class group. By consecutively composing the Dehn twists corresponding to each critical point, we get the identity element of the mapping class group. This sequence of Dehn twists is called a positive factorization and it turns out there is a correspondence between genus g Lefschetz fibrations and positive factorizations of the identity into Dehn twists in the mapping class group of a genus g surface.

By finding new positive factorizations of the identity, one can hope to find interesting constructions of Symplectic 4-manifolds. Genus one Lefschetz fibrations are well understood, this work aims to further explore the realm of genus two Lefschetz fibrations. In particular, my work examines small genus two Lefschetz fibrations which have less than 30 critical points, these are called small. I am working on constructing interesting symplectic 4-manifolds by constructing Lefschetz fibrations through new positive factorizations of the identity in the mapping class group.
 

Problem Keywords 
four dimensional spaces
Scientific Keywords 
Lefschetz fibrations
symplectic manifold

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