Course Control Number: 24846

Instructor: David Nadler

Office Hours: Fridays 3:30-4:30pm, zoom id: 936 2923 9224

GSI: German Stefanich

Office Hours: Fridays 11am-1pm, zoom id: 982 5809 2595

Lectures: Mondays and Wednesdays 3-4:30pm, zoom id: 941 2768 9910, passcode: 264259

Lectures will be a mix of traditional presentations and breakout sessions with group discussion.

Lecture notes:

1. Lecture 1 (W 8/26/20)
2. Lecture 2 (M 8/31/20)
3. Lecture 3 (W 9/2/20)
4. Lecture 4 (W 9/9/20)
5. Lecture 5 (M 9/14/20)
6. Lecture 6 (W 9/16/20)
7. Lecture 7 (M 9/21/20)
8. Lecture 8 (W 9/23/20)
9. Lecture 9 (M 9/28/20)
10. Lecture 10 (W 9/30/20)
11. Lecture 11 (M 10/5/20)
12. Lecture 12 (W 10/7/20)
13. Lecture 13 (M 10/12/20)
14. Lecture 14 (W 10/14/20)
15. Lecture 15 (M 10/19/20)
16. Lecture 16 (W 10/21/20)
17. Lecture 17 (M 10/26/20)
18. Lecture 18 (W 10/28/20)
19. Lecture 19 (M 11/2/20)
20. Lecture 20 (W 11/4/20)
21. Lecture 21 (M 11/9/20)
22. Lecture 22 (M 11/16/20)
23. Lecture 23 (W 11/18/20)
24. Lecture 24 (M 11/23/20)
25. Lecture 25 (M 11/30/20)
26. Lecture 26 (W 12/2/20)

Question? Try asking on Piazza.

Prerequisites: Familiarity with point-set topology and abstract algebra.

Primary source: A. Hatcher, Algebraic Topology, available here.

Syllabus: Fundamental group and covering spaces, simplicial and singular homology, cohomology and Poincare duality.

Roughly Chapters 0-3 of Hatcher's textbook, covered roughly as follows:

• Week 1 (8/24-8/28): (i) Classes start on Wednesday 8/26; (ii) Ch 0, p1-10 (cell complexes, operations on spaces).
• Week 2 (8/31-9/4): (i) Ch 0, p11-17 (criteria for homotopy equivalence); (ii) Ch 1, p21-28 (fundamental group).
• Week 3 (9/7-9/11): (i) No class Monday 9/7: Labor Day; (ii) Ch 1, p29-34 (fundamental group of circle)
• Week 4 (9/14-9/18): (i) Ch 1, p34-38 (functoriality of fundamental group); (ii) Ch 1, p40-46 (van Kampen).
• Week 5 (9/21-9/25): (ii) Ch 1, p46-52 (applications of van Kampen); (ii) Ch 1, p56-62 (covering spaces).
• Week 6 (9/28-10/2): (ii) Ch 1, p63-70 (Galois theory of coverings); (ii) Ch 1, p70-78 (deck transformations).
• Week 7 (10/5-10/9): (ii) Ch1, p87-96 (K(G, 1)); (ii) Ch 2, p97-107 (simplicial homology).
• Week 8 (10/12-10/16): (i) Ch1, p108-113 (singular homology); (ii) Ch 2, p113-131 (structures/properties of singular homology).
• Week 9 (10/19-10/23): (i) Ch 2, p134-137 (degree); (ii) Ch 2, p137-146 (cellular homology).
• Week 10 (10/26-10/30): (i) Ch 2, p146-149 (Euler characteristic, group homology); (ii) Ch 2, p149-153 (Mayer-Vietoris).
• Week 11 (11/2-11/6): (i) Ch 2, p153-155 (coefficients); (ii) Ch 2, p160-165 (Eilenberg-Steenrod axioms).
• Week 12 (11/9-11/13): (i) Ch 2, p166-176 (Hurewicz and classical applications); (ii) No class Wednesday 11/11: Veterans Day.
• Week 13 (11/16-11/20): (i) Ch 3, p185-204 (cohomology of spaces); (ii) Ch 3, p206-214 (cup product).
• Week 14 (11/23-11/27): (i) Ch 3, p214-219 (Kunneth formula); (ii) No class Wednesday 11/25: Thanksgiving week.
• Week 15 (11/30-12/4): (i) Ch 3, p230-249 (Poincare duality); (ii) Ch 3, p249-252 (duality and cup product).

Evaluation: Each week there will be homework. There will also be two midterms and a final exam.

Midterm 1: during Week 6 (9/28-10/2); covering material through end of Week 4 (9/14-9/18).

Midterm 1

Midterm 1 solutions

Midterm 2: during Week 12 (11/9-11/16); covering material through end of Week 9 (10/26-10/30).

Midterm 2

Midterm 2 solutions

Grades will be determined by homework (20%), the midterms (20% each), and the final (40%).

Homework: Unless otherwise noted, problems are from Hatcher's textbook.

Homework is due each Monday at noon. Please submit assignments on Gradescope. Contact the GSI German with any questions.

1. Due Monday 8/31/20: Ch. 0, Ex. 3, 6, 10, 14, 16, 18.
2. Due Monday 9/7/20: Ch. 0, Ex. 19, 23; Ch. 1.1, Ex. 2, 3, 5, 6.
3. Due Monday 9/14/20: Ch. 1.1, Ex. 8, 9, 10, 12, 13, 14.
4. Due Monday 9/21/20: Ch. 1.1, Ex. 16, 18; Ch 1.2, Ex. 3, 4, 10, 14.
5. Due Monday 9/28/20: Ch 1.2, Ex. 16, 21, 22; Ch 1.3, Ex. 4, 5, 7.
6. Due Monday 10/5/20: Ch 1.3, Ex. 8, 10, 11, 14, 16, 18.
7. Due Monday 10/12/20: Ch 1.3, Ex. 23, 32; Ch 1.B, Ex. 2; Ch 2.1, Ex. 2, 5, 9.
8. Due Monday 10/19/20: Ch 2.1, Ex. 17, 20, 21, 22, 23, 29.
9. Due Monday 10/26/20: Ch 2.2, Ex. 1, 2, 7, 8, 9, 11.
10. Due Monday 11/2/20: Ch 2.2, Ex. 16, 17, 18, 21, 24, 29.
11. Due Monday 11/9/20: Ch 2.3, Ex. 1; Ch 2.B, Ex. 1, 2, 3, 4, 8.
12. Due Monday 11/16/20: Ch 2.B, Ex. 10; Ch 2.C, Ex. 4, 5.
13. Due Monday 11/23/20: Ch 3.1, Ex. 3, 4, 5, 6, 11, 13.
14. Due Friday 12/4/20: Ch 3.2, Ex. 4, 8, 12; Ch 3.3, Ex. 6, 7, 8, 24, 26.