Date of Award


Document Type

Honors Thesis (Open Access)


Colby College. Mathematics and Statistics Dept.


Dr. Scott A. Taylor

Second Advisor

Dr. Fernando Q. Gouvêa


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.


Trisection, Heegaard Splitting, Manifold, Handle Decomposition