Math 752 Homepage (Introductory topology II)
Occurs: MWF 8:50-9:40, B313
Van
Vleck.
Instructor: Craig Westerland.
Office Hours: Wednesday
4:30-5:30, Friday 10-12.
Text: Our main text will be
Hatcher's text includes all the algebraic topology that we will cover
in 752. For the differential topology, I recommend
- V. Guillemin and A. Pollack: Differential Topology.
Of course, as with most graduate classes, there is a wealth of other
relevant texts that you may find useful:
- J. R. Munkres, Elements
of Algebraic Topology.
- W. Massey, A Basic
Course in Algebraic Topology.
- J. P. May, A Concise
Course in Algebraic Topology.
- E. Spanier, Algebraic
Topology.
- R. Bott and L. W. Tu: Differential
forms in algebraic topology.
- M. Spivak: Calculus
on Manifolds.
- J. Milnor and J. Stasheff: Characteristic classes.
Description: The first
half of the class will be mostly algebraic topology, as we develop the
homological algebra necessary to study and compute cohomology. The
relevant theorems are the Universal Coefficient and Kunneth theorems.
We then study cup and cap products in (co)homology. The big theorem
here is an application to the study of manifolds: Poincare duality.
This leads us to the more geometric part of the course: we study
differential forms and integration on manifolds. We connect to the
first half of the course by interpreting these constructions in terms
of (deRham) cohomology: cup products of cohomology classes correspond
to wedge products of forms and cap products correspond to integration
(leading to a geometric description of Poincare duality). The deRham
theorem is the major tool for these comparisons.
Finally, we round out the semester with fixed point theorems
(especially the Lefschetz fixed point theorem), and more duality
theorems (e.g. Alexander duality). If there is time, we'll move on to
bundle theory and characteristic classes.
Grading: There will be homework
(collected every week) and a
(take home) final exam given the final week of classes. The
homework will account for 60% of the grade, and the final will account
for 40%.
Homework: All problems are
taken from Hatcher unless otherwise indicated. Note: one should take the problems
from the version of Hatcher that is available online (with errors
corrected).
- (due 2/7) 3.1 #1, 3, 5, 6
- (due 2/14) 3.1 #8, 9, 12
- (due 2/26) 3.2 #2, 3, 5, 7
- (due 3/12) 3.2 #1, 6, 15 (except the part about problems 12, 13,
14), 18
- (due 3/30) 3.3 #4, 7, 8, 9
- (due 4/18) 3.3 #17, 20, 21, 26
- (due 4/27) 3.3 #28, 29; 3.2 #4; 2.C #3
Final Exam: The exam is a
take-home, and due on the last class, Friday, May 11. Here is the
.pdf file.