\magnification=\magstep1
\parskip=5pt
\def\nd{{\noindent}}
\def\P{{\cal{P}}}
\def\R{\mathop{{\rm I}\kern-.2em{\rm R}}\nolimits}
%**end of header
\centerline{\bf Algebraic Topology}
\centerline{Homework 1: Due September 1}      % e.g. September 13
%
%  Now simply type text and problems.  For small print use \small.  To
%  make a problem write
%  \problem ... \endproblem.
%  For a problem with many sections, use
%  \probsec ... \endprobsec.
%  Start each section of the problem with \sec.
%  For a series of true/false questions on a test, use
%  \truefalse ... \endtruefalse
%  Start each problem with \tf.
%  Reset the numbering using \probno=0.

There will be weekly homework assignments due each Wednesday at the {\it
beginning\/} of class. You may study in teams, but it's a good idea to
try the problems on your own first.  Then discuss them
and work together with classmates, friends, pets, favorite oracle,
etc.  In the end, I expect you to write up your own solutions to the
problems.  

These homework assignments will have a large overlap with the homework that 
I assigned last fall, as there are only so many good problems in the book! For
that reason, I have taken down the homework solutions from last year's 
class web page. They'll go back up in December, in time to help you study 
for prelims. I hope to post solutions to this year's problems each week,
shortly after they are due. 
 
Most students will write out their solutions by hand (legibly, please!), but
some may prefer to TeX things. Either way is fine. 

%\flushpar
%{\bf Problems in Guillemin/Pollack}

%\medskip

%\noindent Section 1-2 (page 5):  2, 3, 5

%\bigskip
 
 
{\bf \nd Problem 1}.  Find an example of a topological space $X$ that is 
not Haussdorf such that each point in $X$ has a neighborhood homeomorphic
to the open interval $(-1,1)$. 

{\bf \noindent Problem 2.} Show that the figure 8 (viewed as a subset of 
the plane, with a topology induced from the usual topology of $\R^2$)
is not homeomorphic to a circle. 

{\bf \nd Problem 3.} Let $T_1$ be the surface of revolution obtained
by rotating the circle $(x-2)^2+y^2=1$ around the $y$ axis. Let $T_2$
by the closed unit square with opposite edges identified. (Explicitly,
$T_2 = [0,1] \times [0,1] / \sim$, where $(x,0)\sim (x,1)$ and $(0,y)
\sim (1,y)$ for all $0 \le x \le 1$ and all $0 \le y \le 1$.) Give
$T_1$ the topology inherited from $\R^3$ and give $T_2$ the quotient
topology. Prove that $T_1$ and $T_2$ are homeomorphic.

The point of this exercise isn't to see that the torus is the same thing
as a square with opposite sides identified -- it's to understand the quotient
topology well enough to prove it. 

{\bf \nd Problem 4.} Massey, page 13, exercise 5.1

{\bf \nd Problem 5} Showing that Euclidean spaces aren't homeomorphic can be 
remarkably difficult in higher dimensions. It's easy to see 
that $\R^0$ and $\R^1$ are not
homeomorphic (prove it!), and that $\R^1$ and $\R^2$ are not 
homeomorphic (prove that, too!). But what about $\R^2$ and $\R^3$?  These 
``obviously'' are different, but can you prove it? Discuss how you would
attempt such a proof.  You probably don't have the tools to make a rigorous
proof yet, but I'd like you to write down a few ways in which the spaces
differ, and discuss the challenges of showing that these differences are 
preserved by homeomorphism.  (Eventually we will develop tools, such as
local homology and relative homology, that distinguish 
between $\R^n$ and $\R^m$ for all $n\ne m$.)


\bye
