Monday:
 Study: Read section II.7 and II.8. We'll have very
little to say about II.7 in class, but should look at Theorem
II.7.1 and its (simple) proof carefully. Also read sections
III.13. We'll also have little to say about Chapter III in
class.
 Do:
 Work problems 7.2 and 7.5 (this is identical to 8.2).
 Recall that a map $f:X\to Y$ is nullhomotopic if it is
homotopic to a constant map. We say that $X$ is contractible
if $\operatorname{id}_X$ is nullhomotopic. Note that this is weaker than the
definition given on page 45 of the text. (It is the same as
Definition VII.4.5 on page168.) Prove that the following are
equivalent.
 $X$ is contractible.
 Every map $f:X\to Y$ is null homotopic for every $Y$.
 Every map $g:Y\to X$ is null homotopic for every $Y$.
 Suppose that $f:X\to Y$ has left homotopy inverse $h:Y\to X$
(i.e., $h\circ f\simeq 1_X$) and a right homotopy inverse $g:X\to Y$
(i.e., $f\circ g\simeq 1_Y$. Show that $f$ is a homotopy equivalence.
More generally, show that if $f\circ g$ and $h\circ f$ are homotopy
equivalences, then so is $f$.
 Show that $\mathbf{R}^3\setminus(z\hbox{axis})$ is homotopy equivalent to $S^1$.

Wednesday:
 Study: Read III.16.
 Do:
 Let $\mathsf{Ab}$ be the category of abelian groups with group
homomorphisms. Show that coproducts exist in $\mathsf{Ab}$ and
that the coproduct of $\{G_i,i\in I\}$ is given by the weak
product. (In this case, the weak product is usually denoted
$\bigoplus_{i\in I} G_i$ and is called the direct sum.) In
this problem you can use the facts proved about the weak product
proved in sections III.2, and Theorem II.2.1 in particular.
 Work problems 4.3, 4.5 and 4.8 (assuming 4.7).
 Work problem 3.3 using the fact that a free abelian group on
$S=\{x_i :i\in I\}$ is isomorphic to $\bigoplus_{i\in I} F_i$,
where $F_i$ is the infinite cycylic group generated by $x_i
$.

Thursday (XHOUR):
 Study: Read IV.12
 Do:
 Give a careful statement of the sense in which Theorem IV.2.1
characterizes $\pi(X)$ up to isomorphism.
 Work problems IV.3.1 and IV.3.2. Think about IV.3.3 (more on
this later).

Friday:
 Study: Read IV.35.
 Do:
 Work IV.5.4. (Use an appropriate $n$gon with edge identification as in Example 5.2 in the text).
 Let $X\subset \mathbf{R}^3$ be the union of $n$ (distinct) lines through the origin. Computer $\pi(\mathbf{R}^3\setminus X)$.
 Suppose that $X$ and $Y$ are homotopy equivalent and that $X$ is path connected. Must $Y$ be path connected? Explain.
 (No proofs required.) Let $G$ be a finitely generated abelian
group. Find a space $X$ such that $\pi(X)\cong G$.
 We want to exhibit a path connected compact Hausdorff space $X$
for which $\pi(X)$ is uncountable. Let $C_n$ be the circle in
$\mathbf{R}^2$ of radius $\frac 1n$ centered at $(\frac1n,0)$. Let
$X=\bigcup_{n=1}^\infty C_n$.
 Observe that there is a retraction $r^n$ on $X$ onto $C_n$ given by maping each $C_m$, $m\not=n$, to the base point $(0,0)$. Hence there is a
homomorphism $r^n_*$ of $\pi(X)$ onto $\pi(C_n)\cong
\mathbf{Z}$.
 Let $J_n=[1\frac1n,1\frac1{n+1}]$ and let $h_n:J_n\to I$ be
the unique orientation preserving linear homeomorphism. Let
$f_n:I\to C_n$ be any path in $C_n$. Define $f:I\to X$ by letting
$f_{J_n}=f_n \circ h_n$, and $f(1)=(0,0)$. Show that $f$ is a
path in $X$. (Your proof may include the words "$f$ is clealy
continuous at any $t\not=1$".)
 Show that there is a homomorphism $p$ of $\pi(X)$ onto
$\prod_{i=1}^\infty \mathbf{Z}$. (Since the latter is
uncountable, this means $\pi(X)$ is as well.)
 Does this result contradict the result from problem IV.3.2 which
shows that the fundamental group of a countable wedge product of
circles is the free group, $\mathbb{F}_\infty$, on countably many
generators (which is countable)? Explain.
