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\centerline{\bf Algebraic Topology}
\centerline{Homework 9: Due Wednesday, November 3}      % e.g. September 13
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{\bf Problem 1.} Page 163, problem 2.1

{\bf Problem 2.} Page 166, problem 3.3. The hard part is the last statement.

{\bf Cone operators.}  Let $X$ be a star-like subset of Euclidean
space, meaning that if $x \in X$, then $tx \in X$ for every $t \in [0,1]$. 

As usual, let $C_n(X)$ be generated by the singular $n$-chains on $X$, modulo
the degenerate chains. Let $C_{-1}=\Z$, with the usual augmentation
map $C_0\to C_{-1}$ that counts points. That is, we're computing {\it reduced}
homology.  Consider the {\it cone operator} 
$\phi: C_n \to C_{n+1}$ defined as follows.
If $n\ge 0$ and $T$ is an $n$-cube, let $\phi(T)(x_1,\ldots,x_{n+1})
= x_1T(x_2,\ldots,x_{n+1})$. If $n=-1$, let $\phi(1)$ be the 0-cube that maps
to the origin. 


{\bf Problem 3.} Show that $\partial \circ \phi + \phi \circ \partial$ is 
the identity on $C_n$. Use this fact to prove that the reduced homology of 
$X$ is trivial. 

{\bf Problem 4.} Let $Y$ be any topological space, and let $X$ be the cone 
of $Y$. Modify the construction of Problem 1 to show that the reduced
homology of $X$ is trivial.  

{\bf Problem 5.} More generally, let $X$ be any contractible space. Use
an appropriate cone operator to show that the reduced homology of $X$ is 
trivial.  

Please do problems 3--5 in order.  Don't just do problem 5 and then say
``3 and 4 are special cases''. They are, but the idea is to go from the
concrete and specific to the abstract and general in stages. 

{\bf Problem 6.} As a continuation of last week's assignment on simplicial
homology, see if you can generalize, finding (and proving!) a formula
for the homology of $\Delta^k_0$ (including its faces and sub-faces). 
The key is constructing an appropriate cone operator, albeit in the
category of simplicial complexes and simplicial chains rather that
topological spaces and singular chains. 



\bye
