Date of Award

2022

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Dr. Scott A. Taylor

Second Advisor

Dr. Fernando Q. Gouvêa

Abstract

The decomposition of a topological space into smaller and simpler pieces is useful for understanding the space. In 1898, Poul Heegaard introduced the concept of a Heegaard splitting, which is a bisection of a 3-manifold. Heegaard diagrams, which describe Heegaard splittings combinatorially, have been recognized as a powerful tool for classifying 3-manifolds and producing important invariants of 3-manifolds. Handle decomposition, invented by Stephen Smale in 1962, describes how an n-manifold can be constructed by successively adding handles. In 2012, Gay and Kirby introduced trisections of 4-manifold, which are a four-dimensional analogues of Heegaard splittings in dimension three. Trisection diagrams give a simple way of understanding and studying 4-dimenisonal spaces. This thesis is intended to give a friendly introduction to the analogies between the theory of Heegaard splittings of 3-manifolds and trisections of 4-manifolds. The way we introduce these two decompositions is based on Gay’s “From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4”, which is a summary and expansion on a mini-course given at CIRM in 2018.

Keywords

Trisection, Heegaard Splitting, Manifold, Handle Decomposition

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