Intro to Topology/ Found to Topology Fall 2023

 

Scheduled Meeting Times
Type Time Days Where Date Range Schedule Type  
Face-to-Face Plus 11:00 am - 12:15 pm T Science Math Learning Center 352 Aug 21, 2023 - Dec 16, 2023 Lecture  
Remote Set Day/Time 11:00 am - 12:15 pm R Remote Instruction MS Teams Aug 21, 2023 - Dec 16, 2023 Lecture

 

                                 

Please note the following information. This page will not be updated beyond August 28, 2023. You should join the class MS Team and follow the information there. You need to be registered for the course with a @unm.edu email. Any other email will disable features of Microsoft Teams. I have sent you by email a link to join the class team. Please follow it in order to request access. You can find the link to join the team in Canvas as well. I will drop from the class all students who do not request access to Microsoft Teams by Friday, September 9 or do not come to class during this period. The remote class on Thursday will be through MS Teams. Homework, including submission, and the commucation of all course information will rely on MS Teams and OneNote.

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.

 

Prerequisites: MATH321. Pre- or co requisites: MATH322.

 

Text: 1. Munkres J. Topology (Pearson) 2nd ed 2. Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd

 

 

Office Hours: I will have office hours through Microsoft Teams on M 4pm-5pm and Th 9:30am-10:30am. You can join using the link in Teams or the link that appears in your Outlook or Teams calendar. I will also check regularly the Posts tab in MS Teams for any questions/ comments, etc. You can also stop by my office if you have a question or send me an email to arrange a meeting.

 

Final Exam:TBD, please doublecheck with the final exam schedule. Students having conflicts with this examination schedule must notify the appropriate instructor beforeTBD. Exams will take place in the rooms in which the individual classes have been meeting.

 

 

 Please note the following guidelines for the course:

Attendance: Attendance at UNM is mandatory, see policy.

Grades: The final grade will be determined by homework (25%), two midterms (50%) and a final exam (25%). The Final Exam score will replace the midterm scores that are lower than the Final Exam score.  Although a small curve will be used, 90% , 80% or 70% of the possible maximum points guarantees at least an A, B or C, respectively. The students registered in MATH431 will have a more generous curve.

 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 unless there was an emergency.

 

Accommodation Statement: In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. Accessibility Resources Center (Mesa Vista Hall 2021, 277-3506) provides academic support to students who have disabilities. If you think you need alternative accessible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner.

 

Homework:   Homework is due every Thursday at the beginning of the class in MS Teams. I encourage you to work on the homework with your classmates, but you are required to write up your own solutions in your own words. To help the grader, please write your solutions neatly using correct grammar and mathematical notation (no points will be given for work that the reader cannot follow).   The ten best homework grades will be used in computing the homework score. Please do not turn-in late homework! The syllabus also lists recommended homework problems.  These are NOT to be handed in. Work as many as it takes for you to understand the material.  You should see me as early and as often as necessary if you are having difficulties with the homework problems .

 

                      

Week

 

Date

Topics/Objectives to be learned

1

Aug. 21

1. Metric spaces - examples, open sets. Closed sets, limits, boundary points, limit points. Completeness, the Baire category theorem.

2

Aug.  28

Separability. Total boundedness. Compactness. Continuous and uniformly continuous functions.

3

Sep. 4 (Labor day)

Product Metric Spaces. Bounded/ Continuous linear operators, Principle of uniform boundedness in a Banach space. Contraction principle.

4

Sep. 11

2. Topological Spaces. Continuous function. Universal properties of quotient and subspace topologies. Uniform limits of continuous functions to a metric space. 

5

Sep. 18

Properties of continuous functions: restrictions, locality, gluing.

Base of a topology. Separation axioms. Urysohn's lemma and Tietze's  extension theorem. 

6

 

Sep. 25

Finish Tietze's extension theorem. Compactness. The Arzela-Ascoli theorem. Locally compact spaces. One point compactification. 

7

Oct. 2

Products of topological spaces - sub-base, base and universal property of the product metric. Separation properties and products. Alexander subbase theorem, Tychonoff's theorem.

8

Oct. 9 

 

Connected, locally connected, path connected and locally path connected spaces.

Fall Break Oct. 12-15

9

Oct. 16

Connectedness and products. Paracompactness. Metrizability.

Exam 1, Thursday Oct. 19.

10

Oct. 23

 3. Topological manifolds. Quotient spaces - compactness, connectedness, separation properties, examples.  Covering spaces. Group action with quotient map a covering.

11

Oct. 30

Uniqueness of lifts. The group of deck transformations; covering actions and the group of deck transformations; examples. Lifting of paths and homotopies to a cover.

12

Nov. 6

4. Homotopy Theory - the fundamental group; induced homomorphism. Homotopic maps. Deformation retracts.

13

Nov. 13

 

Homotopy equivalent spaces. The Monodromy theorem. Fundamental group vs. Aut group of a covering. Brower's theorem. Borsuk-Ulam's theorem.

 

14

Nov. 20

The fundamental group as obstruction to existence of lifts. Classification of covers, coverings and sub-groups of the fundamental group, the universal cover.

 

Thanksgiving

15

Nov. 27

 

Exam 2, Tuesday November 28

The Seiferet - van Kampen theorem. Free products of groups.

16

Dec. 4

Amalgamated product of groups, presentation of groups, and the Seiferet - van Kampen theorem. Examples.  

 

 

 

 

Dec. 11-16 Finals week

Final Exam, Tuesday, December 12, 12:30pm-2:30pm, https://schedule.unm.edu/final-exams/final_exam/fall2023.pdf