Math 225C: Algebraic Topology

Spring 2018

Instructor: Michael Andrews.
Class: MWF 1-1:50pm, MS 5148.
Office hours: Tu 11am-12pm and F 2-3pm, MS 6322.

TA: Kevin Carlson.
Discussion section: Th 1-1:50pm, MS 5137.
Office hours: Th 12-1pm and Th 2-3pm, MS 6160.

Textbook: Hatcher, Algebraic Topology, Chapters 0-3.
Syllabus: CW complexes, the fundamental group, covering spaces, Van Kampen's theorem, singular homology, LESs, excision, the Mayer-Vietoris sequence, axioms for a homology theory, cellular homology, cohomology theory, cup products, Poincare duality, de Rham's theorem.
Prerequisites: Basic knowledge of algebra and point-set topology. Try these problems to check your preparedness.

Some notes I'm using to help prepare my lectures: the fundamental group, homology and cohomology - my Oxford lecture notes, H_n(S^n), H^CW, degree

Homework: Homeworks are due at the beginning of lecture on Monday (weeks 2-5) or Wednesday (weeks 6 onwards). Do not submit homework by e-mail. No late homework will be accepted.
Starred problems will be graded. The others will be checked for completion.

HW Due Problems
1 4/9 0: 1* (using the CW-structure given in class would be useful), 3, 6c (just show the weak deformation retraction from Y to Z), 9*, 10, 11, 16*, 19 (results from chapter 0 and lecture might be helpful), 1.1: 1, 2, 3*, 5*, 6*, 11, 12, 13, 15, 16*, prove pi_1(S^1 v S^2)=Z using thm 4.3 of my notes.
2 4/16 1.2: 3*, 7*, 8*, 11, 14, 16, let X be the space of 1.25; calculate pi_1(X).
3 4/23 1.3: 4*, 6, 9*, 10*, 14*, 17, 18, 32, prove that the map defined in the proof theorem 6.10 (of my notes) is a homeomorphism.
4 4/30 2.1: 3*, 4, 5, 6, 7*, 8, 9*, 11, 12, 14
5 5/9 2.1: 15*, 16*, 17, 18, 22, 26, 29*, 30, 31
6 5/16 2.2: 4, 9, 10, 15 (I already did this in class!)
7 5/23 2.2: 7, 8*, 12, 14, 17, 18*, 19, 20, 21, 22*, 28, 29, 33, (I starred the complement of what I meant to - oh well)
8 5/30 2.2: 40, 41, 43*, 3.1: 1, 2*, 3, 9*, 11, 3.2: 1, 3*, 6, 7*, 9* (I starred the complement of what I meant to - oh well)
9 6/8 3.3 2*, 3*, 6, 7*, 8*, 10, 16, 3.2 10, 11*, 13*, 14, 16, 18

You are encouraged to work in groups on your homework; this is generally beneficial to your understanding and helps you learn how to communicate clearly about mathematics. However, you must write up all solutions yourself. Moreover, since crediting your collaborators is an important element of academic ethics, you should write down with whom you worked at the top of each assignment. You should also cite any sources (other than lectures and the textbook) that you use.

Exams: Final - see end of lecture notes - due 6/15.
The final take-home final exam due on 6/15. Once again, you are free to collaborate and consult textbooks and notes, but you must write up all the solutions yourself, and you must credit your sources and collaborators. Submitting the final exam is mandatory. In particular, note that university policy requires that a student who misses the finals be automatically given F, unless it is due to extreme and documented circumstances, in which case, if the performance in the course is otherwise satisfactory, the grade might be I.

Grading: Numerical grades will be recorded in the MyUCLA gradebook. The composite numerical grade will be computed as 70% HW + 30% Final, and the final letter grades will be assigned based on that. If you believe a problem on a homework or an exam has been graded incorrectly, or that your score was not correctly recorded in the MyUCLA gradebook, you must bring this to the attention of the instructor within 10 calendar days of the due date of the assignment in question, or the date of the exam, and before the end of the quarter (6/15). Grading complaints not initiated within this period of time will not be considered. Please verify in a timely manner that your scores are correctly recorded on MyUCLA.