Course Control Number: 24259

Ed discussion forum: Ask questions here!

Instructor: David Nadler

Office Hours: Tuesdays, 12:30pm, in 815 Evans.

GSI: Ethan Dlugie

Email: dlugie.e@math.berkeley.edu

Office Hours: Mondays, 9AM and Wednesdays, noon, in 853 Evans.

Lectures: Tuesdays and Thursdays, 11am-12:30pm, Hearst Field Annex B1.

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 (8/23-8/27): (i) Classes start on Wednesday 8/25; (ii) Ch 0, p1-10 (cell complexes, operations on spaces).
• Week 2 (8/30-9/3): (i) Ch 0, p11-17 (criteria for homotopy equivalence); (ii) Ch 1, p21-28 (fundamental group).
• Week 3 (9/6-9/10): (i) No class Monday 9/6: Labor Day; (ii) Ch 1, p29-34 (fundamental group of circle)
• Week 4 (9/13-9/17): (i) Ch 1, p34-38 (functoriality of fundamental group); (ii) Ch 1, p40-46 (van Kampen).
• Week 5 (9/20-9/24): (ii) Ch 1, p46-52 (applications of van Kampen); (ii) Ch 1, p56-62 (covering spaces).
• Week 6 (9/27-10/1): (ii) Ch 1, p63-70 (Galois theory of coverings); (ii) Ch 1, p70-78 (deck transformations).
• Week 7 (10/4-10/8): (ii) Ch1, p87-96 (K(G, 1)); (ii) Ch 2, p97-107 (simplicial homology).
• Week 8 (10/11-10/15): (i) Ch1, p108-113 (singular homology); (ii) Ch 2, p113-131 (structures/properties of singular homology).
• Week 9 (10/18-10/22): (i) Ch 2, p134-137 (degree); (ii) Ch 2, p137-146 (cellular homology).
• Week 10 (10/25-10/29): (i) Ch 2, p146-149 (Euler characteristic, group homology); (ii) Ch 2, p149-153 (Mayer-Vietoris).
• Week 11 (11/1-11/5): (i) Ch 2, p153-155 (coefficients); (ii) Ch 2, p160-165 (Eilenberg-Steenrod axioms).
• Week 12 (11/8-11/12): (i) Ch 2, p166-176 (Hurewicz and classical applications); (ii) No class Wednesday 11/11: Veterans Day.
• Week 13 (11/15-11/19): (i) Ch 3, p185-204 (cohomology of spaces); (ii) Ch 3, p206-214 (cup product).
• Week 14 (11/22-11/26): (i) Ch 3, p214-219 (Kunneth formula); (ii) No class Thursday 11/25: Thanksgiving.
• Week 15 (11/29-12/3): (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 (9/27-10/1); covering material through end of Week 4 (9/13-9/17).

Midterm 1

Midterm 1 solutions

Midterm 2: during Week 11 (11/1-11/5); covering material through end of Week 9 (10/18-10/22).

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 11:59pm. Please submit assignments on Gradescope. Contact the GSI with any questions.

1. Due Monday 8/30/20: Ch. 0, Ex. 3, 6, 10, 14, 16, 18.
2. Due Monday 9/6/20: Ch. 0, Ex. 19, 23; Ch. 1.1, Ex. 2, 3, 5, 6.
3. Due Monday 9/13/20: Ch. 1.1, Ex. 8, 9, 10, 12, 13, 14.
4. Due Monday 9/20/20: Ch. 1.1, Ex. 16, 18; Ch 1.2, Ex. 3, 4, 10, 14.
5. Due Monday 9/27/20: Ch 1.2, Ex. 16, 21, 22; Ch 1.3, Ex. 4, 5, 7.
6. Due Monday 10/4/20: Ch 1.3, Ex. 8, 10, 11, 14, 16, 18.
7. Due Monday 10/11/20: Ch 1.3, Ex. 23, 32; Ch 1.B, Ex. 2; Ch 2.1, Ex. 2, 5, 9.
8. Due Monday 10/18/20: Ch 2.1, Ex. 17, 20, 21, 22, 23, 29.
9. Due Monday 10/25/20: Ch 2.2, Ex. 1, 2, 7, 8, 9, 11.
10. Due Monday 11/1/20: Ch 2.2, Ex. 16, 17, 18, 21, 24, 29.
11. Due Monday 11/8/20: Ch 2.3, Ex. 1; Ch 2.B, Ex. 1, 2, 3, 4, 8.
12. Due Monday 11/15/20: Ch 2.B, Ex. 10; Ch 2.C, Ex. 4, 5.
13. Due Monday 11/22/20: Ch 3.1, Ex. 3, 4, 5, 6, 11, 13.
14. Due Friday 12/3/20: Ch 3.2, Ex. 4, 8, 12; Ch 3.3, Ex. 6, 7, 8, 24, 26.