Course Control Number: 23678

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Instructor: Midterm

Office Hours: TBD.

GSI: Tahsin Saffat

Email: tahsin_saffat@berkeley.edu

Office Hours: Mondays 3-5 in 959 Evans and Wednesdays 1-3 in 2 Evans.

Lectures: Tuesdays and Thursdays, 2pm-3:30pm, 740 Evans.

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 (8/22-8/26): (i) Classes start on Wednesday 8/24; (ii) Ch 0, p1-10 (cell complexes, operations on spaces).
• Week 2 (8/29-9/2): (i) Ch 0, p11-17 (criteria for homotopy equivalence); (ii) Ch 1, p21-28 (fundamental group).
• Week 3 (9/5-9/9): (i) Ch 1, p29-34 (fundamental group of circle); (ii) Ch 1, p34-38 (functoriality of fundamental group).
• Week 4 (9/12-9/16): (i) Ch 1, p40-46 (van Kampen); (ii) Ch 1, p46-52 (applications of van Kampen).
• Week 5 (9/19-9/23): (i) Ch 1, p56-62 (covering spaces); (ii) Ch 1, p63-70 (Galois theory of coverings).
• Week 6 (9/26-9/30): (i) Ch 1, p70-78 (deck transformations); (ii) Ch1, p87-96 (K(G, 1)).
• Week 7 (10/3-10/7): (i) Ch 2, p97-107 (simplicial homology); (ii) Ch1, p108-113 (singular homology).
• Week 8 (10/10-10/14): (ii) Ch 2, p113-131 (structures/properties of singular homology); (ii) Ch 2, p134-137 (degree).
• Week 9 (10/17-10/21): (i) Ch 2, p137-146 (cellular homology); (ii) Ch 2, p146-149 (Euler characteristic, group homology).
• Week 10 (10/24-10/28): (i) Ch 2, p149-153 (Mayer-Vietoris); (ii) Ch 2, p153-155 (coefficients).
• Week 11 (10/31-11/4): (i) Ch 2, p160-165 (Eilenberg-Steenrod axioms); (ii) Ch 2, p166-176 (Hurewicz and classical applications).
• Week 12 (11/7-11/11): (i) Ch 3, p185-204 (cohomology of spaces); (ii) Ch 3, p206-214 (cup product).
• Week 13 (11/14-11/18): (i) Ch 3, p214-219 (Kunneth formula); (ii) Ch 3, p230-249 (Poincare duality).
• Week 14 (11/21-11/25): (i) Ch 3, p249-252 (duality and cup product); (ii) No class Thursday 11/24: Thanksgiving.
• Week 15 (11/28-12/2): Catch up + Bonus topics.

Evaluation: Each week there will be homework. There will also be a take-home midterm and a take-home final project.

Midterm: during Week 7 (10/3-10/7); covering material through Week 6 (9/26-9/30).

Midterm

Grades will be determined by homework (20%), the midterm (30%), and the final project (50%).

Homework: Problems are from Hatcher's textbook.

Homework is due each Monday at 11:59pm unless otherwise noted. Please submit assignments on Gradescope. Contact the GSI with any questions.

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