Lecture Room: | B305 Van Vleck Hall |
Lecture Time: | 11:00–11:50 MWF |
Lecturer: | Jean-Luc Thiffeault |
Office: | 503 Van Vleck |
Phone: | (608)263-4089 |
Email: | |
Office Hours: | Right after class or by appointment. |
See the official syllabus.
The textbook for the class is Essential Topology by Martin D. Crossley.
The version I use is the corrected printintg of 2010. For earlier versions, please consult the erratum, especially for homework problems.
Other books that I will draw from (but which you don't necessarily need):
Math 551 and 542, or consent of instructor.
Each two weeks I will assign homework from the textbook (or other sources) and post it here. The homework will be due two weeks later.
Homework 1: (due Feb. 7) 2.7, 3.1, 3.2, 3.3, 3.4, 3.7, 3.8, 3.9, 4.1, 4.2, 4.5, 4.6, 4.8, 4.9.
Homework 2: (due Feb. 21) 5.1, 5.2, 5.4, 5.5, 5.8, 5.10, 6.1, 6.3, 6.5.
Homework 3: (due Mar. 21)
1. Consider a modified, more physical version of Theorem 6.39 (p. 112), as discussed in class, where the vector field vanishes on the boundary instead of being tangent to it. Try to prove or disprove that there is then a zero of the vector field in the interior of the disk, in addition to all the zeros on the boundary.
2. Following exercice 6.6 in the book, prove that the degree of a composition of maps is the product of the degrees (but do it right: the "solution" at the back of the book is useless).
3. Exercises 7.1, 7.2, 7.4 from Crossley.
Homework 4: (due Apr. 4)
1. Derive a normal form for a compact surface with b boundary components. (Modify as needed the proof given in class for surfaces without boundaries, or just quote the results that carry through without modification.)
2. Exercises 8.1, 8.3, 8.5 from Crossley.
Homework 5: (due May 2)
Exercises from Crossley: 9.1, 9.3, 9.4 (read about the universal coeff. theorem), 9.6 (9.7 in some editions: the question starts with "Take the simplicial torus...").
There will be three midterm exams in class, and no final exam. The final grade will be computed according to:
Homework | 40% |
Midterm exam 1 | 20% |
Midterm exam 2 | 20% |
Midterm exam 3 | 20% |
The midterm exams will be given during the regular 50 minute lecture period.
Wednesday February 16, 2011 at 11:00–11:50. | |
Monday March 28, 2011 at 11:00–11:50. | |
Wednesday May 4, 2011 at 11:00–11:50. |
lecture | date | sections | topic |
1 | 01/19 | 2 | Continuity |
2 | 01/21 | 3 | Topological spaces |
3 | 01/28 | 3 | Continuity in subspace topology |
4 | 01/31 | 4.1 | Connectivity |
5 | 02/04 | 4.2–4.3 | Compact and Hausdorff spaces |
6 | 02/07 | 5.1 | Homeomorphisms |
7 | 02/09 | 5.2–5.3 | Disjoint unions and product spaces |
8 | 02/11 | 5.4 | Quotient spaces |
9 | 02/14 | 5.4 | Quotient spaces (cont'd) |
– | 02/16 | – | midterm 1 |
10 | 02/18 | 6.1 | Homotopy |
11 | 02/21 | 6.2 | Homotopy equivalence |
12 | 02/23 | 6.3 | The circle: Path lifting |
13 | 02/25 | 6.3 | The circle: Homotopy lifting |
14 | 02/28 | 6.4 | Brouwer's fixed-point theorem |
15 | 03/02 | 6.5; Hatcher 2.2 | Vector fields |
16 | 03/04 | 7.1 | Simplicial complexes |
17 | 03/07 | 7.2 | Triangulation of surfaces |
18 | 03/09 | 7.2–7.3; Fulton C17 | Classification of surfaces |
19 | 03/11 | 8; Fulton C17 | Classification of surfaces (end); Homotopy group of the circle |
20 | 03/21 | 8.1 | Homotopy groups |
21 | 03/23 | 8.1 | Homotopy groups (cont'd) |
22 | 03/25 | 8.2–8.3 | Induced homomorphisms; Fundamental group |
– | 03/28 | – | midterm 2 |
23 | 03/30 | see notes | Mapping class group of the torus |
24 | 04/01 | see notes | Anosov homeomorphisms |
25 | 04/04 | see notes | From the torus to the sphere |
26 | 04/06 | see notes | Topological stirring (with movies!) |
27 | 04/08 | 8.4 | Path connectivity and π_{0} |
28 | 04/11 | 9.1 | Simplicial homology modulo 2 |
29 | 04/13 | 9.1 | Computing simplicial homology modulo 2 |
30 | 04/15 | 9.2–9.3 | Integral simplicial homology |
31 | 04/18 | Adams 1.1–1.2 | Knot theory |
32 | 04/20 | Adams 1.3–1.4 | Reidemeister moves and links |
33 | 04/22 | Adams 1.5 | Tricolorability |
34 | 04/25 | Adams 2.1–2.2 | Tabulating knots |
35 | 04/27 | Adams 2.3 | Tangles and Conway's notation |
36 | 04/29 | Adams 6.1 | Polynomial invariants |
37 | 05/02 | Adams 7.1 | DNA and knot theory |
– | 05/04 | – | midterm 3 |
– | 05/06 | – | career day |
The lecture notes are available in one large PDF file (11.4MB) or a much smaller DjVu file (2.5MB).