Math 552, Elementary Geometric and Algebraic Topology - Spring 2010

Nigel Boston

Contact Information

Telephone: 263-4753, 265-3817.

E-mail: boston@math.wisc.edu

Homepage

Office Hours: 12-1 Mon and 2:30-3:30 Fri, 303 Van Vleck; 12-1 Tues, 3619 Engineering Hall; or by appointment.

Text

Lecture Outline

1. Jan 20: Introduction.
2. Jan 22: Identification spaces.
3. Jan 25: Examples of identification spaces.
4. Jan 27: Finish 4.2 and start fundamental groups.
5. Jan 29: 5.1 and start of 5.2.
6. Feb 1: Most of 5.2 Construction of the fundamental group.
7. Feb 3: End of 5.2 and start 5.3.
8. Feb 5: Calculation of the fundamental group of S^1.
9. Feb 8: Orbit spaces (4.4), examples, and their fundamental groups.
10. Feb 10: No class.
11. Feb 12: Fundamental group of a product and start 5.4 (homotopy type).
12. Feb 15: Rest of 5.4.
13. Feb 17: Brouwer fixed-point theorem. 5.5.
14. Feb 19: Separation of the plane. 5.6.
15. Feb 22: Went over HW1 solutions. More of 5.6.
16. Feb 24: Rest of 5.6 and 5.7.
17. Feb 26: HW2 solutions. Review for 1st Midterm.
18. Mar 1: 1st Midterm.
19. Mar 3: Triangulations. 6.1.
20. Mar 5: Finish 6.1.
21. Mar 8: Return and go over 1st Midterm. Start 6.2.
22. Mar 10: Finish 6.2 (Barycentric subdivision) and start simplicial approximation 6.3.
23. Mar 12: Finish 6.3 and start 6.4, the edge group of a complex.
24. Mar 15: Generators and relations of G(K,L), examples.
25. Mar 17: Fundamental group of Klein bottle, proof that G(K,L) is pi_1.
26. Mar 19: Proved van Kampen's theorem and gave applications.
27. Mar 22: HW3 solutions. Started surfaces, Euler characteristics (7.3).
28. Mar 24: Properties of Euler characteristic, start surgery (7.4).
29. Mar 26: Finish surgery, show every closed surface homeo to known one.
30. Apr 5: 7.5 Surface symbols - that no standard closed surfaces are homeo to each other.
31. Apr 7: 8.2 Introduction to simplicial homology.
32. Apr 9: 8.3 Examples of homology groups.
33. Apr 12: HW4 solutions.
34. Apr 14: That the abelianization of pi_1 is H_1.
35. Apr 16: 8.4 Simplicial maps and homology.
36. Apr 18: Review for Midterm.
37. Apr 21: 2nd Midterm.
38. Apr 23: 8.5 Stellar subdivision.
39. Apr 26: Jordan Ellenberg subbed.
40. Apr 28: No class.
41. Apr 30: 8.6 Invariance.
42. May 3: 9.1 Maps of spheres (degree and applications).
43. May 5: Hairy ball theorem and 9.2 Euler-Poincare formula.
44. May 7: 9.3 Lefschetz fixed-point formula.

Sections

• Main Lecture: MWF 11:00-12:00, B123 Van Vleck.

Course Description

We will cover as much of Armstrong as we can, starting with Chapter 4.
1. Fundamental group and applications: classification of closed 2-manifolds, elementary homotopy theory, the fundamental group of the circle, covering spaces.
2. Simplicial homology: simplexes, triangulation, homology groups, Euler characteristic, simplicial approximation.
3. Selected topics: fixed point theorems, singular homology, knot theory, group actions.

Homework Assignments

Homework will be set every other Wednesday and due two weeks later. You are encouraged to discuss the exercises with your classmates but the work you hand in should be your own. You will be expected to read the sections of the book to be covered in advance of class.

1. Due Feb 12: Chapter 4: 1,2,3,4,5,7. Chapter 5: 1,2.
2. Due Feb 24: Chapter 5: 6,7,9,12,15,17,25,26.
3. Due Mar 10: Chapter 5: 32,33,34,37,40,45,46,49.
4. Due Mar 24: Chapter 6: 1,3,5,10,11,17,18,20.
5. Due Apr 14: Chapter 7: 3,4,5,8,11,12,17,24.
6. Due Apr 30: Chapter 8: 2,8,12,13,16,20,22,25.

Midterms

1st Midterm: Mon, Mar 1. 2nd Midterm: Weds, Apr 21. Both in class.

Final Exam

5:05-7:05pm, Thursday, May 13, in B123 Van Vleck.

Grading Policy