# Columbia University GU4051 section 001 Topology, fall 2022

## Basic information

 Call number: 12826 Room/Time: TuTh 2:40pm--3:55pm, 417 Math Instructor: Mikhail Khovanov Office: 620 Math Office hours: Tentative: Zoom Wednesdays 3-4pm and Office (620 Math) Thursdays 1-2pm E-mail: Graduate Teaching Assistant: Song Yu,   email: Graduate TA office hours: Wed 12-3pm. Sept 14, 21 via Zoom, then in Help Room, 406 Math Undergraduate Teaching Assistant: TBA Undergraduate TA office hours: TBA in Help Room, 406 Math Midterm 1: Thursday, October 6 Midterm 2: (Tentative) Thursday, November 17

Prerequisites: The first semester of "Introduction to Modern Algebra" (Math W4041) or equivalent is strongly advised, since we'll be using basic group theory in the second half of the course. You also need to be familiar with how to write proofs. Familiarity with analysis (Honors Math A,B or Introduction to Modern Analysis) is recommended as well.

Textbook: Topology, by James R. Munkres, second edition. Available on Amazon. We are using the more expensive Pearson edition (listed around \$100 plus tax). Cheaper (and probably identical except for the paper quality) Prentice-Hall edition is available on Amazon as well. To be on the safe side, my recommendation is to get Pearson edition (first link).

Additional resources: A.Hatcher, Notes on Point-Set Topology
A.Hatcher, Algebraic Topology (we discussed coverings of the figure eight graph, see Chapter 1.3 starting on page 56).

Syllabus: Topological spaces and continuous maps. Examples of topologies.
Closed sets, interior, limit points. Hausdorff spaces.
Product topology.
Connectedness and path-connectedness.
Compactness.
Metric spaces and their topology.
Countability and separation axioms
Normal spaces, the Uryson lemma.
Homotopic maps.
The fundamental group. Examples.
Retraction and homotopy equivalence.
Groups via generators and relations.
Free groups and amalgamated products.
The van Kampen theorem and its applications.
Surfaces and their classification.

Lectures:
Lecture 1 (Tu, Sept 6): Topology, basis for a topology (Sections 12, 13). Notes
Lecture 2 (Th, Sept 8): Order topology, product topology, subspace topology (Sections 14, 15, 16) Notes
Lecture 3 (Tu, Sept 13): Closed sets, closure and interior of a set, limit points, Hausdorff spaces (Section 17) Notes
Lecture 4 (Th, Sept 15): Continuous functions, homeomorphisms (Section 18) Notes
Lecture 5 (Tu, Sept 20): Topology on the image of a continuous map, metric spaces
Lecture 6 (Th, Sept 22): More metric spaces, product topology for infinite products (Sections 19,20), Cantor set as an infinite product. Notes
Lecture 7 (Tu, Sept 27): Continue with Sections 19-21. More about box and product topologies. Uniform topology. Convergence of sequences. First-countability of metric spaces. Notes
Lecture 8 (Th, Sept 29): Clarification about uniform topology. Continuity of sums and products of functions. Start on connected spaces (Sections 23, 24). Notes
Lecture 9 (Tu, Oct 4): Continue with connected spaces, connected components, path-connected spaces, path-connected components (Sections 23, 24, 25 (first part, not covering locally-connected spaces). Independent reading: A connected but not path-connected space from the sine curve (Example 7 in Section 24). Notes
Quiz 1    Solutions
Midterm 1 (Th, Oct 6).
Lecture 10 (Tu, Oct 11): Compact spaces I (Sections 26, 27). Notes
Lecture 11 (Th, Oct 13): Compact spaces II (Section 27). Notes
Lecture 12 (Tu, Oct 18): Retract maps and subspaces. Uniform convergence and continuity (end of Section 27 and Theorem 21.6 in Section 21). Various types of compactness (Section 28) Notes

If you want to explore compactness in more depth: Various Forms of Compactness.

Lecture 13 (Th, Oct 20): Compact spaces III: various notions of compactness and their equivalence for metric topologies. Notes. We then discussed Cauchy sequences and complete metric spaces following Hocking and Young. Munkres covers complete metric spaces in a different way in Section 43.
Lecture 14 (Tu, Oct 25) The Peano curve (Munkres, Section 44) Notes
Lecture 15 (Th, Oct 27) Homotopies and path homotopies (Munkres, Section 51) Notes
Lecture 16 (Tu, Nov 1) Fundamental group and covering spaces (Munkres, Sections 52, 53). Quotient topology notes (Munkres, Section 22)
Lecture 17 (Th, Nov 3) Covering spaces (Sections 53, 54). Notes. More about quotient topology (Section 22).
Lecture 18 (Th, Nov 10) by Song You. Covering spaces summary, Brauer fixed point theorem (Section 55) Notes.
Lecture 19 (Tu, Nov 15, on Zoom - video via Courseworks) Proof of Brauer's theorem. Covering spaces of the figure eight. Quotient topology examples. Notes.
Lecture 20 (Th, Nov 17) Continue with covering spaces of the figure eight (also see Hatcher, Algebraic Topology) and free groups. Covering space approach identifies elements of a free group. A criterion for the map to be a quotient map. The Fundamental Theorem of Algebra (Munkres Section 56).
Lecture 22 (Tu, Nov 22) Deformation retracts and homotopy type (Munkres Section 58).
Lecture 23 (Th, Dec 1) Maps of fundamental groups and homotopies. Homotopy equivalent spaces have isomorphic fundamental groups (Munkres Section 58). Fundamental group of the n-sphere (Munkres Section 59). Free product of two groups and its size (discussed in Munkres in greater generality, Section 68).

Homework: Homework will be assigned on Thursdays, due Thursday evening the next week, via an online upload on Courseworks. The first problem set is due September 15. The lowest homework score will be dropped.
The numerical grade for the course will be the following linear combination:
22% homework, 20% each midterm, 35% final, 3% quizzes.

Homework 1 Due September 15.

Homework 2 Due September 22.

Homework 3 Due September 24.

Homework 4 Due October 7.

Homework 5 Due October 20.

Homework 6 Due October 27.

Homework 7 Due November 4.

Homework 8 Due November 11.

Homework 9 Due November 25.