Homeworks

  • HW 10, due Monday December 13 by NOON, in my mailbox (16th floor):
    Hatcher Section 1.3: #3, 4, 9, 12, 15


  • HW 9, due Friday December 3 in class:
    Hatcher Section 1.2: #4, 7, 9, 14, 17
    Some remarks.
    For #4: Parts of this can be a bit hand-wavy.
    For #7: What sort of cell complex for S^2 ``induces" a cell complex for the quotient space?
    For #9: A better hint than the one Hatcher gives is that and group homomorphism to an abelian group contains the commutators in its kernel.
    For #17: This does not appear to use section 1.2 material. A simpe but useful observation: Let A = R^2\Q^2 be a subset of X = R^2\q where q is a point in Q^2. Then two loops in A which are NOT homotopic as loops in X, are also not homotopic as loops in A.

  • HW 8, due Friday November 19 in class:
    HATCHER Section 1.1: #3, 5, 7, 12, 16
    Some remarks.
    For #5: It might be useful to consider the quotient space (S^1 times [0,1])/(S^1 times {0})
    For #7: This phenomenon is known as a "Dehn twist" which is a crucial object in studying dynamics on surfaces, as well as low-dimensional topology.
    For #12: If you haven't had group theory, ask me what all the homomorphisms are from the integers to the integers.
    For #16d: the wedge "v" of two spaces, X v Y, is discussed on p.10


  • HW 7, due Friday November 5 in class:
    Section 43: #5, 6a,b,c, 8
    Section 45: #4, 5
    Hints on how to start off with the hint for 45.5:
    Let Y be the one-point compactification of X. Y exist? Define the "extend" function E :C_0(X,R) -> C(Y,R) by g = E(f) where g(x) = f(x) and g(infty) = 0. Does it really map into C(Y,R)? Show it's an isometry. Exercise 28.6 implies it's a homeomorphism. Use Exercise 43.8 somehow to how the image of E is closed. Ascoli Theorem then appears somehow.


  • HW 6, due Friday October 29 in class. DELAYED UNTIL MONDAY:
    Section 28: #1
    Section 29: #3, 8
    Extra problem: Let X be a compact Hausdorff space, and let A be a closed subset of X. Show that the quotient space X/A is homeomorphic to the one-point compactification of X-A.


  • HW 5, FRIDAY October 22 in class:
    Section 26: #7,8,10a,12
    Section 27: #2, 3
    Extra Section 27 problem: Give an example of a closed and bounded set in a metric space which is NOT compact.


  • HW 4, due Weds October 13 in class:
    Section 23: #4, 11, 12
    Section 24: #2, 3, 10
    Section 25: #1, 4 (not 5 as I had initially typed).


  • HW 3, due Monday October 4 in class:
    Section 20: #4, 5, 6 part a only, 8 parts a and b only
    Section 21: #6, 8
    Section 22: #2, 3, 6,


  • HW 2, due Monday September 27 in class:
    Section 18:#8,11,12,13
    Section 19: #3,7,8


  • HW 1, due Monday September 20 in class:
    Section 13:#6,8
    Section 16:#4,10
    Section 17:#6,11,13,14
    For people that want to get ahead, Section 18:#8,11,12,13 will be assigned for HW 2.


  • HW 0, over the first week: read Munkres Sections 1-7 on your own. We will not cover this in the class. Sections 6 and 7 will not be needed for a while. Hand in a sheet describing your experience with other relevant math courses (algebra, topology, analysis) if any.