Math 525 (Rezk, 12:00 MWF, 441 Altgeld Hall).
Here is the syllabus.
References
The main text is Hatcher.
There are other references available online, such as:
Additional material
I'm going to post sections of my lecture notes from time to time, once
I decide they are not nonsense. Note that these are written for my
benefit, and sometimes are a bit cryptic or omit details.
- Notes for Week 1. Review of
topological spaces. Constructions. Compactness and connectedness.
- Notes for Week 2. Hausdorff
property. Paths. Definition of fundamental group and groupoid.
Path components.
Dependence on basepoint. Fundamental group of circle.
- Notes for Week 3.
Winding number. Base point dependence. Induced homomorphisms and
homotopies. No retract theorem. Brouwer fixed point theorem.
- Notes for Week 4.
Borsuk-Ulam theorem. Fundamental theorem of algebra. Fundamental
groups of spheres. Covering maps. Properly discontinuous group
actions. Homotopy lifting property.
- Notes for Week 5.
Covering maps and fundamental group. Monodromy. Action of
fundamental group on fiber of covering. Universal cover.
Construction of universal cover, idea. Local path connectedness.
Semi-local simply connectedness.
- Notes for Week 6.
Construction of universal cover. Construction of covers with
prescribed monodromy local system. Local systems for path connected spaces.
Free products and restricted
free products of groups. Statement of Van Kampen theorem.
- Notes for Week 7.
Proof of Van Kampen theorem. Homotopy extension property. Attaching
cells, and effect on fundamental group.
- Notes for Week 8.
CW complexes. Fundamental group of 1-dimensional CW-complexes. Deck
transformations.
- Notes for Week 9.
Coverings and subgroups. Calculation of the group of deck
transformations. The category of covers, and equivalence with local
systems. Introduction to homology. Delta-complexes.
- Notes for Week 10.
Simplical homology. Chain maps and chain homotopies. Boundary
homomorphism and long exact sequences.
- Notes for Week 11. Singular homology.
- Notes for Week 12. Homotopy
and excision axioms. 5-lemma.
- Notes for Week 13. Homology
of spheres. Good pairs. Reduced homology. Equivalence of
simplicial and singular homology. Mayer-Vietoris.
- Notes for Week 14. No retract
and Brower fixed point theorems in arbitrary dimension. Invariance
of dimension. Jordan curve theorem and generalizations. Degree.
- Notes for Week 15. Local
degree. Cellular homology.
Test
Midterm in class Wednesday, April 4.
Final, Wednesday, May 9.
Final
The final is in class, at a time to be announced. It will be based
on previous comps exams for Math 525. Old
comps exams can be
found here.
(Before 2005, the course was called "Math 430". Older exams may
have questions about de Rham cohomology of subsets of the plane;
this topic isn't in the syllabus anymore, so you can ignore these.)
Homework
- Homework 1. Revised (prob 2a) Jan 22.
Due Monday,
January 23.
- Homework 2.
Due Monday,
January 30.
- Homework 3. Due Wednesday, February
8. (Changed!)
- Homework 4. Due Friday, February
17. (Date changed!)
- Homework 5. Due Wednesday, February
29. (Date changed; I added a clarifying comment to 2(d).)
- Homework 6. Due Friday, March 9.
- Homework 7. Due Friday, March 16.
- Homework 8. Due Friday, March 30.
- Homework 9. Due Monday, April 9
(date corrected).
- Homework 10. Due Wednesday, April
18. (due date changed) Note: the versions of the Axioms should be
the ones I gave in class; I've summarised
them here.
- Homework 11. Due Friday, April 27.
Last modified 4 May 2012 by Charles Rezk. Email: 