Math 6441 Home Page

Math 6441 Algebraic Topology Spring 2022 -	-----------------------------------A.T. Fomenko

Class Meetings

Monday, Wednesday, Friday, 9:30-10:20 am, Skiles 171. Masks are expected in class.

Book

Notes

The lectures will be mainly based on these notes.

Office Hours

Fri 11-12 (Teams) and by appointment

Homework

There will be weekly homework assignments, generally assigned on Fridays. The due date is generally the following Friday at the start of the class. Late homework will be generally accepted through the weekend.

Exam

There will be a closed book, take home midterm during March 4-13. This will cover CW complexes, fundamental groups, covering spaces, and the basics of homology.

Final project

The final project is a writing assignment on any topic that is related to the course and not covered in the course. A (not exhaustive) list of ideas can be found below. Topics, along with at least two references are due March 31. First drafts are due April 17. Each student will give comments on at least three other drafts by April 25. The target due date for final drafts is April 29. All submissions will be posted on Teams in the appropriate channels.

Grading

Course grades will be determined by homework scores (60%), the midterm (20%), and the final project (20%). The lowest homework grade will be dropped. Standard cutoffs for letter grades apply, although these cutoffs may be lowered as warranted.

Weekly Schedule

Week	Dates	Topics	Text	Homework	Lecture notes
1	Jan 10	Intro / Spaces	Chapter 0	p. 18 #1,14,16,20,23	Jan 10 Jan 12 Jan 14
2	Jan 17	Fundamental group	Section 1.1	p. 38 #5,6,A.1	Jan 19 Jan 21
3	Jan 24	Van Kampen's theorem	Section 1.2	p. 38 #7,9,10,12,16	Jan 24 Jan 26 Jan 28
4	Jan 31	Covering spaces	Section 1.3	p. 52 #4,8,9,16, p.79 #10	Jan 31 Feb 2 Feb 4
5	Feb 7	Covering spaces	Section 1.3	p. 79 #4,9,12,14,18	Feb 7 Feb 9 Feb 11
6	Feb 14	Homology	Section 2.1	p. 96 #2, p. 131 #4,5,6,9, and A.2	Feb 14 Feb 16 Feb 18
7	Feb 21	Relative homology	Section 2.1	15,16,17,18,19	Feb 21 Feb 23 Feb 25
8	Feb 28	Excision / Mayer-Vietoris	Section 2.2	Midterm	Feb 28 Mar 2 Mar 4
9	Mar 7	Applications	Section 2.B	None	Mar 7 Mar 9 Mar 11
10	Mar 14	Cohomology	Section 3.1	p. 204 #5,6 & A.3	Mar 14 Mar 16 Mar 18
Spring Break
11	Mar 28	Cup product	Section 3.2	p. 228 #1,3,7	Mar 28 Mar 30 Apr 1
12	Apr 4	Poincaré duality	Section 3.3 Cup & Cap	p. 257 #24 (orientable case),25,26	Apr 4 Apr 6 Apr 8
13	Apr 11	Spectral sequences	Spectral Sequences		Apr11 Apr 13 Apr15
14	Apr 18	Spectral sequences	Spectral Sequences	A.4, A.5 (optional)	Apr 18 Apr 20 Apr 22
15	Apr 25	Spectral sequences	Spectral Sequences		Apr 25

Additional homework

Prove that a contractible space is simply connected.
Summarize the proof of Proposition 1B.9.
Compute, directly from the definitions, the cohomology of M_g, the closed, connected, orientable surface of genus g.
Compute the homology of the torus using the one-at-a-time spectral sequence.
Compute the homology of the Klein bottle using the one-at-a-time spectral sequence.