Date of Award
Honors Thesis (Open Access)
Colby College. Mathematics and Statistics Dept.
Dr. Scott A. Taylor
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
Recommended CitationZhang, Suixin "Cindy", "Decomposing Manifolds in Low-dimensions: from Heegaard Splittings to Trisections" (2022). Honors Theses. Paper 1392.