MAT 530 Topology, Geometry I, Fall 2015.

  • References :
    O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev, Elementary Topology. Problem Textbook.
    A version of this book is available for free download from Viro's webpage. (Caution: the link goes directly to a large pdf file.) This is an unusual textbook: all theorems are presented in step-by-step questions that you have to figure out yourself.
    Note that this online version does not have any solutions; you can buy a complete book, but I don't recommend using solutions too much. (You will need to be able to produce detailed proofs in quizzes! So please work through the questions carefully.)
    The first few weeks of the course will be "flipped": you will be responsible for learning/reviewing the theory via the homework problems; in class, we will address questions and discuss some of the finer points.

    Allen Hatcher, Algebraic Topology.
    This book is available for free download from Hatcher's webpage. (This book will mostly be used for the fundamental group & coverings part. This is Chapter 1 of Hatcher.)

    Other useful books:

    James R. Munkres, Topology.

    William S. Massey, Algebraic Topology: An Introduction, GTM 56. (The first few chapters are also contained in another book by Massey, GTM 127).

  • Exams: Final Exam will be on Monday, Dec 14, 10am-12:30pm, in P-131.
  • Homework: weekly assignments will be posted on this page. Homework will make a significant part of your course grade. There will also be regular quizzes (base on homework material), especially during the first part of the course (point-set topology).

    Important: Please write up your solutions neatly, be sure to put your name on them and staple all pages. Illegible homework will not be graded. Late homework will not be accepted. You are welcome to collaborate with others and even to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.

    There will be no quizzes until further notice.

    Homework 13: pdf, due Wednesday, Dec 2. Please read about deck transformations and group actions in Hatcher p.70-74; we'll go over this topic on Monday.

    Homework 12: pdf, due Wednesday, Nov 18.

    Homework 11: Hatcher p. 53 questions 3, 4, 8, 9. Also, compute the fundamental group of the projective plane and of the Klein bottle by using van Kampen theorem: represent the projective plane as a Mobius band with its boundary glued up by a disk, the Klein bottle as the union of two Mobius bands glued along their boundary circles. Hand in this part, due Wednesday, November 11.
    Go through (at least the first part of) 37'4x of Viro's book to learn about winding number.
    Cellular spaces: read 42'1 in Viro's book, do 42.1, 42.2, 42.G, 42.H (do not hand in). Do 43.Ax, 43.Bx, 43.Ex, 43.Fx, 43.Hx, 43.Ix, 43.Jx; you may assume that all the cellular complexes are finite (except for the questions such as 43.Ix that become vacuous if you assume finiteness). Hand in 43.Ax, 43.Bx, 43.Ix only

    Homework 10: Hatcher p. 39 question 16 (except (f), which we did in class); p. 79 question 4 (guess what the covering should look like, and prove that it works); Viro 34.M, 34.21, 34.24, 35.7, 40.A. Hand in on Wednesday, Nov 4.
    Please also go through Viro 34'1- 34'5, especially 34'3, which we haven't discussed in class. (Do not hand in anything except the questions listed above, feel free to skip questions such as 34.9 and 34.23.)

    Homework 9: Hatcher p. 18 questions 1, 10; p. 39 questions 10,14, 15, and two more questions pdf . Hand in on Wednesday,Oct 28.
    Optional: question 6 p. 18; you will need question 5 for the proof (do not hand in).

    Homework 8: section 1.1 of Hatcher's book, questions 3, 5, 7, 13 on p. 38 (to hand in on Wed, Oct 21). Caution: questions 5 and 7 deal with the general notion of homotopy, not homotopy of paths. This material is contained in Chapter 0 of Hatcher and will be discussed in class on Monday.
    Reading: "Homotopy and homotopy type" in Hatcher Chapter 0; section 1.1 of Hatcher EXCEPT "The fundamental group of the circle". Sections 30-33 in Viro's book are also useful (but are not a mandatory assignment).
    Please also do take-home make-up, due Monday, Oct 19. Please carefully write the solutions you need to improve/redo on separate paper (do not correct the original exam!) Hand in your make-up solutions together with your exam. While reworking your solutions, you are allowed to use books and notes to better understand the material.

    Homework 7:
    Part I (do not hand in) Elem. Top. Problems: 23 A-H, I-L (basic material on projective spaces) to be done by Monday
    25'1x-25'4x (lettered quesions only, skip numbers; skip 25.Hx if unfamiliar with complete metric spaces)
    Section 25 is about various topologies on spaces for maps X --> Y. Some of it will be familiar from analysis; compact-open topology is important in topology and geometry.
    Part II (hand in, due Wednesday, October 14) 25.Cx, 25.Ix, 25.Lx; 22.26. 22.O. For 22.O, you might need a lemma: any homeomorphism of the sphere Sn-1 can be extended to a homeomorphism of the entire disk Dn (we are thinking of the sphere as the boundary of the disk). Prove this lemma first (sometimes it's called the Alexander trick).
    Note that the lemma is breaks down if you want to get a smooth (i.e. differentiable) map; in fact, Milnor's famous exotic 7-dimensional spheres are obtained by gluing two copies of D7. The resulting manifolds are homeomorphic but not diffeomorphic to the standard 7-sphere!

    Homework 6:
    Part I (do not hand in) Elem. Top. Problems: 19'1x (19.1x, 19.2x, 19.Dx, 19.Ex, 19.Fx only); 19'2x;
    20'3-20'5 (set-theoretic summaries 20'1 and 20'2 may be useful!); 20'6 (20.26-20.29 only); 20'7 (do letters N-U, skip numbers 20.37-20.42); 20.8;
    21'1-21'3, 21'4-21'5 (only as much as you need for 22'1), 22'1-22'3.
    Also, take a look at further examples of quotient spaces in 22'5, 22'6, 22'8, 22'9-22'13 (no need to prove them carefully).
    We will be discussing the remaining questions on compactness + product spaces on Monday, quotient spaces on Wednesday.
    Part II (hand in, due Monday, October 5) pdf

    Homework 5:
    Part I (do not hand in) Elem. Top. Problems:
    17'1, 17'3-17'5, 17'6 (skip 17.11, 17.12), 17'7, 17'8 (skip 17.17-17.23), 17'9 (X,Y,Z only); 16'2 (skip 16.11), 16'3-16'6; 18'1, 18'2 (skip 18.2)
    We will discuss this material in the same order (section 17 first, then section 16, then section 18).
    Section 16 requires familiarity with countable sets; 16'1 contains everything you need to know. For a quick and fun refresher, you can watch a video about the Infinite Hotel and a video about countable and uncountable sets.
    Part II (hand in, due Wed 9/23) 18.3x-18.5x. (If you are not familiar with normed spaces, just think about the corresponding metric d(x, y) = ||x-y|| = sup|xn - yn|. By definition, this will give the same topological space.)

    Homework 4:
    Part I (do not hand in) Elem. Top. Problems: 13'1-13'2, 13'3x (13.6x and 13.7x only); 14'1-14'7 to be done by Monday, Sept 14, and 15'1-15'5, 15'7-15'8 to be done by Wednesday, Sept 16. (We will be discussing (path-)connectedness in Monday class and separation axioms on Wednesday.) There will be a quiz on Wednesday as usual.
    Part II (hand in, due Wed 9/16) 14.27x (1), (2) only; 14.28x (1),(2) only.

    Homework 3:
    Part I (do not hand in) Elem. Top. Problems: 12'1- 12'8 (except question 12.31), to be done by Wed 9/9.
    Part II (hand in, due Wed 9/9) 11.23 (please try to write a reasonably careful proof, although no formulas are required)
    10.6 (please also give an example where the inclusion is strict)
    3'5 (3.9 and 3.10) Use topology to show that there are infinitely many primes. (Give an "honest" new topological proof as suggested in 3'5; rewriting Euclid's proof in topological terms will be worth little credit.) Arithmetic progression is a sequence of the form {am+b}, m=0,1,2,..., where a and b are positive integers. Hint: suppose that the set of primes is finite. Show that {1} would then be open.
    Please also do these questions about the Cantor set, a remarkable nowhere dense set.

    Homework 2:
    Part I (do not hand in) Elem. Top. Problems: 3'6, 4'11, 4'12, 4'14, 5'1-5'3, 6'1-6'7, 6'12 (6.M, 6.N, 6.32, 6.33 only), 6'13 (6.O, 6.37-6.40, 6.42 only); 6'14, 10'1-10'8, 10'10 (10.O, 10.P only), to be (mostly) done by Monday, Aug 31. Don't forget to do both "theorems" (labeled by letters, eg 6.B), and "examples" (labeled by numbers, eg 6.19).
    Added on 08/31, to be done by 09/2: 11'1-11'9: please do all the lettered questions 11.A-11.Y, numbered questions 11.1-11.8, 11.10, 11.19, 11.20 only, and (optional) any of questions 11.21-30 that seem interesting or difficult to you.
    We'll be discussing this material in class on Monday and Wednesday. (There will be a quiz on Wednesday, Sept 2.)
    Part II (hand in, due Wed 9/2) 2'Ix (note that this about a disjoint union -- the question is not trivial), 4'15x (4.Mx-4.Px only). Please write complete proofs.

    Homework 1: Elem. Top. Problems: 2'1-2'10; 3'1-3'3; 4'1-4'5; 4'9-4'10, to be done by Wednesday, Aug 26.

  • Syllabus: we will follow the basic outline from the graduate core course requirements (see below), not necessarily in the same order, with additional topics as time permits.
    1. Basic point set topology
      • Metric Spaces
      • Topological spaces and continuous maps
      • Comparison of topologies
      • Separation axioms and limits
      • Countability axioms, the Urysohn metrization theorem
      • Compactness and paracompactness, the Tychonoff theorem
      • Connectedness
      • Product spaces
      • Function spaces and their topologies, Ascoli's theorem
    2. Introduction to algebraic topology
      • Fundamental group
      • Fundamental group of Sn; examples of fundamental groups of surfaces
      • Seifert-van Kampen theorem
      • Classification of covering spaces, universal covering spaces; examples
      • Homotopy; essential and inessential maps

    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.