# Topology (Math 1410)

Undergraduate course, *Brown University*, 2020

## Course summary

This is an undergraduate course in topology, which studies properties of geometric objects. In this course we will learn

- basic principles of topology (e.g. topological invariants, point-set notions);
- proofs of famous classical results (Heine-Borel, van Kampen, Brouwer fixed point, classification of surfaces);
- connections of topology to other areas of math (free groups, fundamental theorem of algebra) and to the world (robots, hanging pictures, car shades);
- how to calculate topological invariants such as Euler number and fundamental group;
- how to read/write proofs and critique arguments, especially those that are based on pictures/visualization.

## Textbook

Armstrong, *Basic topology*

## Topic schedule

The lectures for the course were pre-recorded, and are available on YouTube. Start here.

- Week 1: Introduction. (Armstrong Chapter 1, Sections 1-2)
- Week 2: Topological spaces. (Armstrong Chapter 1)
- Week 3: Point-set topology. (Armstrong Chapter 2-3)
- Week 4: Compactness and connectedness. (Armstrong Chapter 3)
- Week 5: Quotient spaces. (Armstrong Chapter 4)
- Week 6: Fundamental group. (Armstrong Chapter 5)
- Week 7:
- Week 8:
- Week 9: Triangulations and the fundamental group (Armstrong Chapter 6)
- Week 10: van Kampen for triangulated spaces
- Week 11: Classification of surfaces. (Armstrong Chapter 7)
- Week 12: (Thanksgiving week)
- Lecture 32: Proof of the classification theorem slides
- Lecture 33: Surfaces with boundary (video, see slides from Lecture 32)

Week 13: Reading period

- Week 14: Exam week (submit final project)

## Discussion sessions

Discussions sessions happen Monday 5-6pm and Tuesday 12-1pm. You only need to attend one (they are the same). This is an opportunity to get live practice and feedback. It is recommended that you are up-to-date on lectures and the reading before the discussion session.

Week 2 (9/14): cutting and pasting with surfaces

Week 3 (9/21): topologies on a set with 3 elements

Week 4 (9/28): closure and complement on subsets of ℝ

- Week 5 (10/5): Conway-Coxeter Frieze patterns
Week 6 (10/12): no discussion (university holiday)

Week 7 (10/19): Midterm review

Week 8 (10/26): Hexiamonds

Week 9 (11/2): no discussion (election day)

Week 10 (11/9): Sprouts

Week 11 (11/16): 4-color theorem

- Week 12 (11/23): magic car shade