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.