# Math 552: Elementary Geometric and Algebraic Topology

### Spring 2022 Course Information

Class schedule/Location: MWF 12:05 PM -12:55 PM in Van Vleck B119
Office: 625 Van Vleck Hall
Email: bhepler [AT] wisc [DOT] edu
Office hours: MF 2-3pm, or by appointment. Virtual office hours TBD.
Canvas page: https://canvas.wisc.edu/courses/285013

Textbook: Topology (2nd edition) by James Munkres (amazon link).
Occasionally, we’ll use Algebraic Topology by Allen Hatcher as a supplement. Hatcher’s book is freely available on his webpage, and you can get a physical copy quite cheaply via amazon if you’d like (not required).

Grade: your course grade will be comprised of: homework (30%), one midterm (30%), and a final exam (40%). The midterm will be held during our usual class time, some time in late February/early March (TBD). The final will be a take-home exam.

Homework: will be assigned regularly and will be the main source of problems appearing in the exams. I’ll post the assignments on this page, in addition to our course canvas page. Once the HW is posted, you’ll have one week to complete it and hand it in. Late HW will not be accepted.

### Course description

1. Fundamental Groups and Covering Spaces:
The first part of the course is a brief introduction to the methods of Algebraic and Geometric Topology. It starts with Poincaré’s definition of the fundamental group of a space, and various methods to compute it, such as the Seifert-van Kampen theorem. It proceeds with the classification of surfaces, and a detailed study of covering spaces. Applications include the Brouwer fixed point theorem, and the Borsuk-Ulam theorem.
2. Simplicial Complexes and Simplicial Homology:
Time permitting, the second part of the course is a brief introduction to the methods of Combinatorial Topology and Homological Algebra. It starts with simplicial complexes and their realizations, and proceeds to simplicial homology groups, and ways to compute them. We will illustrate these techniques with concrete examples, and derive some applications.