Available at: https://digitalcommons.calpoly.edu/theses/3236
Date of Award
3-2026
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Patrick Orson
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
One studies the symmetries of an algebraic object by looking at its automorphism group. For example, the automorphism group of a regular n-gon is generated by rotations and reflections, and the automorphism group of {1,...,n} is generated by transpositions. We study the group of symmetries of a closed, oriented 2- and 3-dimensional manifold M, called the mapping class group of M. In the 2-dimensional case, it is known explicitly what the generators of the mapping class group are. We present a proof of this fact following Farb and Margalit's exposition. The question about the mapping class group of 3-manifolds is currently not fully answered. However, 3-manifolds admit a "nice" decomposition into handlebodies, which are easier to work with. We examine the mapping class group of a 3-manifold M by considering its action on the Heegaard splitting of M. More specifically, we show that a finite subgroup of the mapping class group of M that fixes its Heegaard splitting can be realized as the deck transformation group of some finite-sheeted covering space of M.