MAT 364 Topology and Geometry, Fall 2010.

The course will cover a range of topics. We will look at some weird pictures and play with the famous Mobius band and Klein bottle. We'll also make a connection to analysis classes and work with limits and continuous functions. We'll finally learn what those epsilon and delta really mean!

As this is an upper-level class, familiarity with proofs is expected. Indeed, you will have to write careful proofs in your homework.

  • References :
    L. Christine Kinsey, Topology of surfaces.
    This is a required text, although we won't always follow it section by section.

  • Exams: there will be two midterm exams and a final exam.

    Final Exam: Wednesday, Dec 15, 5:15-7:45pm in P-116. (Our usual room.) The final exam is cumulative and covers everything we studied during semester.

    Midterm I: Wednesday, Oct 13, in class.

    Midterm II: Friday, Nov 19, in class. Note change of date!! Checklist of topics for exam II is here. The exam focuses on material covered since the first midterm; although you will not be tested on the Exam I material, you are still responsible for basic concepts learned earlier in the semester.

    The first exam covers the material we've learned so far. (including things that are not in the textbook.) A checklist of topics is here.

  • Homework: weekly assignments will be posted on this page. Homework will constitute a significant part of your course grade.

    Important: For each homework problem, please give a proof or detailed explanation as appropriate (unless otherwise stated). Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with others and to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.

    Week 1 (08/30 – 09/03) sections 1.1, 2.1 (up to Definition 2.10).
    Homework 1, due Sept 8: please do the following exercises from the book.
    1.1 (use your intuition about continuous deformations; no explanations required).
    2.1 and 2.3 for the sets E, F and G only (i.e. parts 4,5 and 6). In addition, find isolated points for each of these sets. Explain your answers.
    2.6, 2.9, 2.11 (give proofs). Some solutions
    Also, determine which of the following identities are true:
    (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
    (A ∩ B) ∪ C = (A ∩ C) ∪ (B ∩ C)
    (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C). Give proofs or counterexamples to support your answer.

    Week 2 (09/08) section 2.1 (more on open and closed sets)
    Homework 2, due Sept 15: please do the following exercises from the book.
    2.7, 2.8, 2.12, 2.15, 2.16. Prove everything. Some solutions

    Week 3 (09/14 – 09/17) sections 2.2, 2.3
    Homework 3, due Sept 22: Do the following exercises from the book, as well as two additional questions below. Prove everything.
    2.19, 2.20, 2.22, 2.25, 2.26 ab. Some solutions
    -- Give an example of sets A, B, such that B consists of a single point, B ⊂ A ⊂ R2, B ≠ A, and B is both open and closed relative to A.
    -- Which of the following are true? Here, f : X → Y is a function, A, B subsets of X, C, D subsets of Y, and f -1 (C) stands for pre-image of the set (not for the inverse function, which might not exist.)
    f (A ∩ B) = f (A) ∩ f (B)
    f -1 (C ∩ D) = f -1( C) ∩ f -1 (D)
    f -1 (C ∪ D) = f -1 (C) ∪ f -1 (D). Prove or give counterexamples to support your answers.

    Week 4 (09/20 – 09/24) sections 2.3, 2.5
    Homework 4, due Sept 29: pdf Some solutions

    Week 5 (09/27 – 09/31) sections 2.5, 2.6
    Homework 5, due Oct 6: pdf Solutions

    Week 6 (10/3 – 10/8) section 2.4 (+extra material)
    Homework 6, due Oct 13: pdf Solutions

    Week 8 (10/18 – 10/22) section 3.2, part of section 3.1 (we took Thm 3.5 as a definition of topology)
    Homework 7, due Oct 27: pdf Solutions

    Week 9 (10/25 – 10/29) Metric space and their topology
    Homework 8, due Nov 3: pdf Solutions

    Week 10 (11/1 – 11/5) Surfaces: 4.1, start 4.3
    Homework 9, due Nov 10: pdf Solutions

    Week 11 (11/8 – 11/12) Classification of surfaces: 4.3, 4.5
    Homework 10, due Nov 17: pdf Solutions

    Exam on Nov 19

    Week 12 (11/21 – 11/23) Idea of the Euler characteristic (Chapter 5). Platonic solids (pp.105-107)

    Week 13 (11/29 – 12/3) The Euler characteristic, topological invariance (5.2-5.4)
    Homework 11, due Friday, Dec 3 : pdf

    Homework 12, due Friday, Dec 10 : pdf

    Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.