Topology II Course Information
Department of Mathematics University of Cincinnati
Topology II (15-Math-605-001) Winter Quarter 2004
Instructor Prof. David A. Herron (813 Old Chem Bldg, 556-4075) Office Hours: M,W,F 10-11
Textbook Topology (second edition) by Munkres
General Syllabus Chapters 4,6,7,9,11
This page is a work in progress!
Below I list information regarding: this week's hot topics , suggested problems, homework.
5-9 Jan
12-16 Jan
21-23 Jan This week we'll finish paracompactness and begin chpt 7. Our goal is to understand the compact-open topology in function spaces. I suggest reviewing:
- section 20: the uniform metric; theorems 20.4,20.5; problems 4-8 (especially 6)
- section 45: theorem 45.1
- section 43: 43.1,43.2,43.3,43.4, the uniform metric
26-28 Jan Well, thanks to the snow day on Monday we didn't finish the compact-open stuff. Oh well.
2-6 Feb This week we'll finish chpt 7 and begin our excursion into the realm of algebraic topology! Now is a good time to review all that point-set topology in preparation for the upcoming midterm. Here are some ideas to help....
- p.289:#11 looks like a good problem to help you better understand the various product/function space topologies
- p.280:#7 is a great "metric space" workout; if you like it, try #8 too
- there are some interesting supplementary exercises on pp.317-318; also pp.228-229
9-13 Feb Midterm exam time!
16-20 Feb
This week we delve ever deeper into the mysteries surrounding the fundamental group. In regards to problem 3 (p.335 in Munkres) I'd like you to also/first answer the following:
- When will a given pair of paths joining two points induce the same isomorphism of the fundamental groups?
- When will all paths joining two points give induce the same isomorphism of the fundamental groups?
23-27 Feb
This week we start learning about covering spaces. We study the lifting problem and prove the Unique Lift Theorem, the Path Lifting Theorem, and the Path Homotopy Lifting Theorem.
1-5 Mar
This week we finally compute the fundamental group of the circle. Yahoo! Then we study deformation retracts and homotopy equivalence.
8-12 Mar
During our last week we investigate the Seifert-Van Kampen Theorem and see how to compute the fundamental groups of some special surfaces.
Here are suggested problems for each indicated section.
- 29, Local Compactness, p.186: 1,2,3,5,6,8,10,11
- 30, Countability Axioms, pp.194-195: 1-5,9-17
- 31, Separation Axioms, p.199: 1-8
- 32, Normal Spaces, pp.205-207: 1-5, 9
- 33, Urysohn's Lemma, pp.212-213: 1-9
- 35, Tietze's Extension Theorem, pp.223-224: 1-9
- 36, Embedding manifolds, p.227: 1-5
- Supplementary Exercises, pp.228-229: lotsa good stuff here
- 39, Local Finiteness, p.248: 1-6
- 41, ParaCompactness, pp.260-261: 1-7,10
- 20, Metric Topology, pp.126-128: 4-8,11
- 43, Complete Metric Spaces, pp.270-271: 1-4,7-9
- 45, Compactness in Metric Spaces, pp.280-281: 1-8
- 46, Compact-Open Topology, pp.288-289: 1-6(routine),7,89,11
- Supplementary Exercises, pp.317-318: lotsa good stuff here,1-5,7,8?,9
- 51, Homotopy, p.330: 1-3
- 52, Fundamental Group, pp.354-355: 1,2,3,4,5,6
- 53, Intro to Covering Spaces, p.341: 1-6 all good
- 54, ULT, PLT, PHLT, pp.347-348: 1-3(do!),4,5,6-8 good too
- 55,56,57, Applications
- 58, Def Retracts and Homo Equiv, pp.366-367: 1&3(easy),2(fun!),4(do it!),5&6(easy),7,8
- 59, Fundamental Group of the Sphere, p.370: 1-2(good),3(easy),4good
- 60, Fundamental Groups of some Surfaces, p.375: 1-5 all good
Here is the assigned homework with due dates.
| Due Dates | Page:Problem |
|---|
| 7,9 Jan |
|
186:3 |
186:10 |
| 12,14,16 Jan |
194:15 |
213:5 |
223:5 |
| 21,23 Jan |
|
260:6 |
227:5 |
| 26,28 Jan |
224:8 |
260:8 |
|
| 4,6 Feb |
|
289:7 |
289:8 |
| 9,11,13 Feb |
330:1 |
Midterm |
330:3 |
| 16,18,20 Feb |
|
|
335:5 |
| 23,25,27 Feb |
335:3 |
335:6 |
341:6b |
| 1,3,5 Mar |
348:4,5 |
348:8 |
366:4 |
| 8,10,12 Mar |
366:7 |
366:8 |
370:4 |
| 17 March |
Final Exam 8-10 |