Manifolds Final Solutions
1. a) Its party time! X is an innite string of balloons, or more precisely the real line
with a sphere attached to each integer point. The covering map is just the usual exponential
map to S 1 on the real line, and the identity m
Manifolds Midterm solutions
Inclass problems:
1. By assumption 1X is homotopic to a constant map cx0 . Hence f = f 1X is homotopic
to f cx0 by a lemma from text, done.
2. a) Suppose : X Y is a map. Dene : (X )(Y ) by ([x]) = [(x)], where
[x] denotes the
Homework 2 and Homework 1 Solutions
NOTE: Since the actual writing up of proofs is a very timeconsuming process, occasionally I will assign problems that are highly recommended but not to be written up or
turned in. Be sure to at least understand the res
Homework 9 Solutions to Selected Problems
2. Consider the diagram of pointed spaces
npp
X X p p p p p X
q
q q
?
?
GG
m

G
If there is a map n making X a topological group and q a continuous group homomorphism, then it is uniquely dened by the above comm
Homework 5 Solutions
1. Let H (x, t) = (1 t)f (x) + tg (x). Since f and g are never antipodal, this homotopy
takes values in Rn+1 0. So H/H  is the desired homotopy.
2. Problem 7.3. The assumption X pathconnected is not needed, since all the action is
Homework 7, due Friday Nov. 16
Reading: Chapter 9.
Homework: This is one problem, The Example of Great Enlightenment, divided into
several parts. It will be continued in future assignments.
1. Show that the following three groups are isomorphic:
a) The in
Homework 8, due Wednesday Nov. 21
Reading: Chapter 10, but keeping in mind the following:
1. In class Ill give a somewhat more economical statement and proof of Seifertvan
Kampen, emphasizing the concept of pushout. You could omit these parts of the text
Homework 8 Solutions
1. a) It is trivially checked that X M or (X, Z ) is a contravariant functor CSets.
Hence it takes isomorphisms to bijections, proving the only if. Conversely, suppose for
all objects Z , the natural map M or (Y, Z )M or (X, Z ) given