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.