Intro to Topology/ Found to Topology
Instructor: Dimiter Vassilev Office: SMLC 326 Email: vassilev@unm.edu Phone: 505 277 2136
Math 431 Introduction to Topology & Math 535 Foundations of Topology
Description: Metric spaces, topological spaces, continuity, algebraic topology. Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces.
Prerequisite: MATH401
Text: Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd edition
Please
note the following guidelines for the course:
Grades: The final grade will be determined by homework & quizzes
(25%), two midterms (50%) and a final exam (25%).
Exam scores are posted on
https://learn.unm.edu/ .
Homework: Homework from the previous week is due Monday at the beginning of class. There will be one HW
weekly. You can work together on the homework, but you do need to
write up your own solutions in your own words. To help the grader, please write
your solutions up neatly and clearly (no points for work that the reader
cannot follow- this is also true for exams), and staple the sheets.
The best ten homework/quiz grades only will be counted for a total of
100pts. Please no late homework
Missed
Exams: Make-up exams can be
arranged for exams missed with a VALID excuse (illness, family emergency,
active participation in scholarly or athletic activities), and ONLY if prior
notice is given.
Disability
Statement:
We will accommodate students with documented disabilities. During the first two
weeks of the semester, those students should inform the instructor of their
particular needs and they should also contact Accessibility Services in Mesa
Vista Hall, Room 2021, phone 277-3506. In addition, they should see CATS-
Counseling and Therapy Services;
Homework – Fall 2015 (please check after class as advanced postings of homework could change)
Turn in only the odd numbered problems.
| Week |
Date |
Topics |
Odd numbered Homework Problems are due the first class of the following week |
| 1 | Aug. 17 | 1. Metric Spaces - completeness, the Baire category theorem | HW1 |
| 2 | Aug. 24 | Compactness - spaces of continuous functions. | HW2 |
| 3 | Aug. 31 |
Principle of uniform boundedness in a Banach space. Contraction principle. |
HW3 |
| 4 | Sep. 7 | 2. Topological Spaces | HW4 |
| 5 | Sep. 14 | Continuous function. Base of a topology. | HW5 |
| 6 | Sep. 21 |
Separation axioms. Urysohn's lemma and Tietze's extension theorem. Compactness. The Arzela-Ascoli theorem. |
HW6 |
| 7 | Sep. 28 | Connected and path connected space. Locally compact spaces. | HW7 |
| 8 |
Oct. 5
October 8–9 |
Exam 1, Wednesday Oct. 7 Fall Break |
Exam scores are posted on https://learn.unm.edu/ |
| 9 | Oct. 12 | Products of topological spaces - Alexander subbase theorem, Tychonoff's theorem. | HW8 |
| 10 | Oct. 19 |
3. Quotient spaces, group actions and covering spaces - Hausdorffness of the quotient space; group action with quotient map a covering; uniqueness of lifts. |
HW9 |
| 11 | Oct. 26 |
The group of deck transformations; the quotient under the group of deck transformations; examples |
HW10 |
| 12 | Nov. 2 | HW11 | |
| 13 | Nov. 9 |
4. Homotopy Theory - the fundamental group. Free products of groups. The Seiferet - van Kampen theorem. |
HW12 |
| 14 | Nov. 16 |
The Seiferet - van Kampen theorem. Deformation retracts. Homotpic maps and homotopy equivalent spaces, consequences for the fundamental groups. Brower'r theorem. Covering spaces- lifting of paths and homotpies. The Monodromy theorem. Fundamental group vs. Aut group of a covering. |
HW13 |
| 15 |
Nov. 23 November 26-27 |
Exam 2, Wednesday Nov. 25 Thanksgiving |
|
| 16 | Nov. 30 | Borsuk-Ulam's theorem | HW14 |
| Dec. 7 Finals week | Wednesday, December 9, 3:00pm - 5:00pm | ||