Course: | MTH 3105 (Topology I) |
Instructor: | Alex Suciu |
Time and Place: | Mon. & Wed., 5:30 - 7:00 PM, in 544 NI |
Office Hours: | Mon. & Wed., 4:00 - 5:00 PM, in 441 LA |
Prerequisites: | MTH 3010 (Basics of Analysis) |
Textbook: | Basic Topology, by M. A. Armstrong, Springer-Verlag, UTM, corrected 4th printing, 1994 |
Grade: | Based on problem sets, class projects, and a final exam |
Continuity, Compactness and Connectedness | ||
Identification Spaces | ||
The Fundamental Group | ||
Triangulations | ||
Surfaces | ||
Simplicial Homology | ||
Degree and Lefschetz Number | ||
Knots and Covering Spaces |
The first (shorter) part of the course treats General Topology. The objects of study are metric and topological spaces. The main properties that are studied are connectedness and compactness. We also introduce several constructions of spaces, and study the invariance of various properties under topological equivalence.
The second part of the course treats the basics of Algebraic Topology. It starts with the fundamental group of a space, and methods to compute it. It proceeds with a study of simplicial complexes, and the classification of surfaces. Simplicial homology is then developed. Applications include the Brouwer fixed point theorem, the Euler-Poincaré formula, the Borsuk-Ulam theorem, and the Lefschetz fixed point theorem. (A more thorough treatment of some of these topics may be postponed for Topology II.)
The last part of the course serves as a brief introduction to Geometric Topology. It starts with covering space theory, and the correspondence between coverings of a space and the subgroups of the fundamental group of that space. It ends with a brief excursion into Knot Theory: the fundamental group of a knot complement, Seifert surfaces, and the Alexander polynomial.
Here are some past qualifying exams in Topology, based in large part on the material covered in this course.
Chapter | Page | Problems | |
---|---|---|---|
Homework 1 | 2 | 31 | 2, 3, 4 |
2 | 35 | 17 | |
2 | 36 | 20 | |
3 | 50 | 7, 11, 18 | |
Homework 2 | 3 | 55 | 23, 25 |
3 | 63 | 43 | |
3 | 72-73 | 2, 3, 5, 10 | |
Homework 3 | 4 | 78 | 16, 21 |
4 | 85 | 26, 27, 32 | |
5 | 91 | 5, 7 | |
Homework 4 | 5 | 95 | 11, 13 |
5 | 102 | 21 | |
5 | 109 | 27, 28, 31 | |
Homework 5 | 6 | 124 | 8 |
6 | 131 | 14, 17 | |
6 | 140 | 20, 22, 23, 24 | |
Homework 6 | 7 | 170 | 28, 29, 30 |
8 | 183 | 11, 16, 17 | |
8 | 184 | 18, 19 |
Department of Mathematics | Office: | 441 Lake Hall | Messages: | (617) 373-2450 |
Northeastern University | Phone: | (617) 373-4456 | Fax: | (617) 373-5658 |
Boston, MA, 02115 | Email: | alexsuciu@neu.edu | Directions |
Created: August 10, 1998 Last modified: Sept. 20, 1999
URL: http://www.math.neu.edu/~suciu/mth3105/top1.f98.html |