Math 307, Fall 2010
Homework 10, due Friday, December 3
Note: In problems (2) and (4), some kind of pictorial representation of the triangulation is all that is
required.
(1) Find the fundamental group of the spaces from problem 4 on Homework 9.
(2) (p.12
Math 307, Fall 2010
Homework 8, due Friday, November 12
(1) (p.102, 1) Use Theorem 5.13 to show that the Mbius strip and the cylinder both have fundamental group
o
Z.
(2) Let p and q be relatively prime integers (not necessarily prime).
(a) Show that S 3
Math 307, Fall 2008
Homework 4, due Friday, October 8
(1) (p.50, 14) Let f be a one-to-one map from a compact space X to a Hausdor space Y . Show that f is an
embedding of X into Y , i.e. f : X f (X) is a homeomorphism.
(2) (p.50, 15): A space is locally
Math 307, Fall 2010
Homework 6, due Friday, October 29
(1) (p.78, 16) Show that O(n) and SO(n) Z2 are homeomorphic. (It turns out they are also isomorphic as
groups for n odd, but not for n even as they have centers of dierent sizes; center of O(n) has tw
Math 307, Fall 2008
Homework 3, due Friday, October 1
(1) (p.22, 10) Find a homeomorphism from the real line to the interval (0, 1) and show any two open intervals
are homeomorphic.
(2) (p.22, 11) Each of the following pairs of spaces can be continuously
Math 307, Fall 2010
Homework 2, due Friday, September 24
(1) What are the limit points of Q R? (This should be a one-line answer; its ok to use something you know
from analysis about how the rationals distribute among all the reals.)
(2) (p.31, 1(a)(b)(c)
Math 307, Fall 2010
Homework 7, due Friday, November 5
(1) (p.91, 1) Let S 1 denote the unit circle in R2 as usual. Suppose f : S 1 S 1 is a map which is not homotopic
to the identity map. Show that f (x) = x for some point x of S 1 .
(2) A space X is con