# MAT 530: Geometry and Topology I (Fall 2022)

### About the course

The field of topology
was developed about 100 years ago to provide foundations for analysis and
geometry. In the course, we will look at the most important definitions and
results from basic point set topology and elementary algebraic topology. For
textbook, we will use the second edition of *Topology* by James Munkres;
the paperback version costs around $40 on amazon. (Here is a link to an online copy.)
Please have a look at the syllabus for additional
information. When I taught the course in 2014, I typed up lecture notes; you can find those here.

### Time and location

We meet on Tuesday and Thursday, 9:45–11:05 am, in room **P-117**
of the Physics building. There is an extra (inofficial) recitation hour for
interested parties, on Friday, 2–3 pm (in our usual classroom
**P-117**). Other office hours are on Monday, 2:00–4:00pm in room
**4-110** of the Mathematics building.

### Midterm

The midterm was held in class on Tuesday, October 18. It was based on the lecture notes (1–13) and the homework (1–6). Here are the solutions. The average was 22 (out of 50) and the median 27; the highest score was 49, the lowest one 0.

### Make-up lecture

There will be a make-up lecture on Tuesday, December 6, at 9:45 in our regular classroom P–117. The topic is applications of homology groups (such as the Jordan curve theorem).

### Homework assignments

- Week 2 (due September 6)
- Week 3 (due September 13)
- Week 4 (due September 20)
- Week 5 (due September 27)
- Week 6 (due October 4)
- Week 7 (due October 13)
- Week 8 (due October 20)
- Week 10 (due November 1)
- Week 11 (due November 8)
- Week 12 (due November 15)
- Week 13 (due November 22)
- Week 15 (due December 6)

### Lecture notes

Here are notes for the lectures so far, together with a list of topics.

- Lecture 1 (August 23): Definition of metric spaces and topological spaces
- Lecture 2 (August 30): Examples of topological spaces
- Lecture 3 (September 1): Closed sets, limit points, continuous functions
- Lecture 4 (September 6): Homeomorphisms, manifolds, quotient and product topology
- Lecture 5 (September 8): Connectedness and path connectedness
- Lecture 6 (September 13): Connected components, definition of compactness
- Lecture 7 (September 15): Properties of compact spaces, subspaces of Euclidean space
- Lecture 8 (September 20): Baire's theorem, locally compact spaces, one-point compactification
- Lecture 9 (September 22): Zorn's lemma, Tychonoff's theorem, connected sums
- Lecture 10 (September 27): Countability and separation axioms
- Lecture 11 (September 29): Urysohn's lemma
- Lecture 12 (October 4): Urysohn's metrization theorem, partitions of unity
- Lecture 13 (October 6): Embedding compact manifolds into Euclidean space, Tietze's extension theorem
- Lecture 14 (October 13): Complete metric spaces, topologies on the space of functions
- Lecture 15 (October 20): Equicontinuity, Ascoli's theorem
- Lecture 16 (October 25): Paths and homotopy, fundamental group
- Lecture 17 (October 27): Basic properties, covering spaces, lifting paths and homotopies
- Lecture 18 (November 1): Examples of fundamental groups
- Lecture 19 (November 3): Deformation retracts, free products of groups
- Lecture 20 (November 8): The Seifert-van Kampen theorem
- Lecture 21 (November 10): Fundamental groups of surfaces, classification of surfaces
- Lecture 22 (November 15): Lifting lemma, covering spaces and subgroups
- Lecture 23 (November 17): Existence of covering spaces
- Lecture 24 (November 22): Deck transformations and regular covering spaces
- Lecture 25 (November 29): Homology and its properties, homology groups of spheres
- Lecture 26 (December 1): Definition of singular homology, some proofs

This note contains an elementary proof for the invariance of domain (and Brouwer's fixed point theorem). It uses Sperner's lemma and general topology, but no algebraic topology; I learned some of the details from Terry Tao's blog.

This note contains an elementary proof of the Jordan curve theorem. It uses some ideas from the proof of the Seifert-van Kampen theorem.