Introduction to Topology: MATH-4040-01

Fall 2012

Instructor: Victor Roytburd (Amos Eaton 405, Ph. 276-6889, roytbv@rpi.edu)

Tentative Office hours: Tu, Fr, 4-5:30 (or by appointment)

Class meets: Tu, Fr 2:00-3:50, Room: Eaton 216

Web-page: http://homepages.rpi.edu/~roytbv//topology/Topology12.htm

Textbook: Colin Adams and Robert Franzosa, Introduction to Topology: Pure and Applied, Prentice Hall. ISBN-13: 978-0131848696. There is a list of corrections and clarifications to the book on-line.

Course Philosophy

Topology is concerned with geometrical properties that are preserved under continuous deformations of objects. These properties are determined only by positioning of points with respect to each other and not by the distances between them (topology's maiden name is Analysis Situs, which means analysis of place). One of the first problems that could be called topological is the Euler's Koenigsberg bridges problem. Topology may become very abstract; in this course the emphasis will be on the geometric and visual underpinnings of topological concepts and results, and on mathematical rigor. One of the goals of the course is helping students to refine their capacities of reading abstract mathematical texts and develop a basic view of topology and its applications.

 

I don't have a rigid agenda as to how much material to cover. The pace of the course will be mostly determined by our progress, which will be measured by the performance on the homework assignments and tests. I am somewhat skeptical about benefits of doing homework problems in groups; and there are some educational research findings that suggest that students working individually master the material better than those working in groups. What is definitely helpful is telling your solutions to your friends for "quality control."

Homework: There will be weekly homework assignments that will be collected and graded. As a rule, homework assignments will be due on Tuesday.

Tests: There will be two tests and a final.

Grading: Homework 35%, Tests 40%, Final 25%

 

Tentative Course Outline

 

I expect to cover the core material of the first seven chapters of the textbook. The really interesting stuff starts after the core chapters; beyond the basics I plan to discuss one of the three topics listed at the bottom the list of the key topics:

 

1.      Topological spaces, bases.

2.      Interior, closure, and boundary.

3.      Subspaces, product spaces, quotient spaces.

4.      Continuity and homeomorphisms.

5.      Metric spaces.

6.      Connected spaces.

7.      Compactness through coverings and limit points; compactifications.

8.      Homotopy and degree of mappings; the Brouwer fixed point theorem.

9.      Knots.

10.  Classification of surfaces (any surface is a spheres with handles), Euler characteristic.

 

If you have any preferences as to which topics out of the last three on the list to consider in the course, please email me.

 

Student Learning Outcomes

This is a list from the Digital Measures course profile.

-         The students will refine their capacities of reading abstract mathematical texts and develop facilities to write brief mathematical proofs.

-         The students will develop a broad view of basic topology and of some applications.

-         The students will develop understanding of basic structure and properties of topological spaces.

-         The students will learn some fundamental properties of continuous mappings of topological spaces.

-         The students will be exposed to some important classes of spaces, such as compact and connected ones.

Statement of Academic Integrity

 

The guiding principle is that work that you present for grading as your own should in fact BE your own. The following statement on Academic Integrity is from the course profile on Digital Measures:

 

Student-teacher relationships are built on trust. For example, students must trust that teachers have made appropriate decisions about the structure and content of the courses they teach, and teachers must trust that the assignments that students turn in are their own. Acts, which violate this trust, undermine the educational process. The Rensselaer Handbook of Student Rights and Responsibilities define various forms of Academic Dishonesty and you should make yourself familiar with these. In this class, all assignments that are turned in for a grade must represent the student's own work. In cases where help was received, or teamwork was allowed, a notation on the assignment should indicate your collaboration. Submission of any assignment that is in violation of this policy may result in a penalty of a grade of F. If you have any question concerning this policy before submitting an assignment, please ask for clarification.

There is a list of corrections and clarifications to the book on-line.

Homework Assignments

Assignment #1, due September 7 One of the problems in the first assignment (Problem 1.34) was incorrect in the first printing of the Textbook. I was not aware of this because in the 2nd printing that I have the error was corrected. My apologies

Assignment #2, due September 14. Answers

Assignment #3, due September 21. Some solutions

Assignment #4, due September 28

Assignment #5, due October 5. Solutions

There will be no class on Tuesday October 9 (Monday, Columbus Day schedule)

Assignment #6, due Tuesday October 16

Assignment #7, due Friday November 2

There will be no class on Friday November 9

Assignment #8, due Friday November 30

Week-by-week schedule

week

Plans

Reading

10/16, 10/19

Review for the test; if any time is left we will proceed with compactness. Test #1.

1.1-1.3, 2.1-2.3, 3.1-3.4, 4.1-4.2, 5.1-5.3, 6.1-6.3

10/23

We will finish the topic of compactness by discussing compactness in metric spaces and one-point compactification. The next topic is the fundamental group π1(X).

7.2, 7.3, 7.5

10/26, 10/30

Circle functions, degree and retraction. Sudden cardiac arrest. The fundamental theorem of algebra.

9.2, 9.3

11/2

Vector fields. Index of a singularity. Vector fields on a sphere. Covering spaces and lifting.

9.5, 9.6

11/06

Lifting. Applications of fixed point theorems to economics

10.1, 10.2

11/09

NO CLASS

 

11/13

The Brouwer Fixed Point Theorem

10.1

11/16

Equilibrium prices

10.2

11/20

Limit point compactness. The closed graph theorem.

 

 

Set-valued functions, the Kakutani fixed point theorem, Nash Equilibrium.

 

 

Manifolds, triangulations, Euler’s characteristic

 

12/4

Normal polygons for surfaces. Homology groups and Betti numbers

 

12/7

Jared Salvadore will give a short presentation on mesh generation for finite elements. Julienne LaChance will talk about D. Gekhtman's work on pursuit dynamics of ensembles of points.

 

12/14

FINAL EXAM: 3-6pm RICKETTS 211

(Mandatory for the students who submitted fewer than 75% of homework papers.)

 

 

Topics for Presentations in Class:

1.     The Alexander Horned Sphere (pp. 339-340 of the textbook)

2.     The Limit point compactness theorem's proof (Theorem 7.36 and related material)

3.     The Ham-Sandwich Theorem

4.     Turning the sphere S2 inside-out

Please let me know if you are interested in giving a presentation (probably about 20-30 minutes long) on any of these topics or any other topological topic of your choice.

Tests

Test #1 will be given on Friday October 19 in class. The test will include topics covered by the first five homework assignments. Namely,

1.      Topological spaces, bases.

2.      Interior, closure, and boundary. Limit points.

3.      Subspaces, product spaces, quotient spaces.

4.      Continuity and homeomorphisms.

5.      Metric spaces.

6.      Connected spaces, not including path connectedness.

 

Test questions will be similar to the shorter homework questions from the homework assignments 1-5. You will be allowed the use of one sheet of hand-written notes.

 

The advanced grade is determined by the performance on the homework assignments (65% of the grade) and the test (35%). The final exam is optional; it will include the material covered by the homework exercises.

 

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Last updated 3 December 2012