Introduction to Combinatorial Topology

Department: 
Math
Course Number: 
4803-LOV
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Not regularly scheduled

Special topics course on Combinatorial Topology, offered in Spring 2022 by Marissa Loving.

Prerequisites: 

MATH 2106 

Course Text: 

I am not planning to have a required textbook for this class, but will share relevant chapters from the Primer, Office Hours with a Geometric Group Theorist, and Topology through Inquiry. I plan to assign regular homework, along with in-class group work, and a final group project with a final presentation and written report. 

 

Topic Outline: 

This course is an intuitive, example-driven introduction to low dimensional manifolds and some combinatorial methods in geometric group theory (focused on the mapping class group and the curve complex), intended to expose students to ideas and proofs in visual mathematics. Examples of low-dimensional manifolds include knots (like a tangled-up string with the ends fused together, which are 1--dimensional) and surfaces (like the peel of an orange or the glaze of a donut, which are 2--dimensional). Furthermore, there are many ways to build 2--dimensional spaces from 1--dimensional spaces, and 3--dimensional spaces from 2--dimensional spaces. This allows us to gain intuition about complicated objects from simpler pieces that we can understand and even draw pictures of!

Topic goals:

1. Topological equivalence

2. Orientability

3. Topology of surfaces

4. Mapping class groups + curve complexes 

5. Examples of 3-manifolds (handlebodies and mapping tori)