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\centerline{\bf Algebraic Topology}
\centerline{Homework 8: Due Wednesday, October 27}      % e.g. September 13
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Most of this week's problem set is about simplicial homology, first of
triangulated surfaces, and then of simplicial complexes in general. But
first some leftover problems on covering spaces:

{\bf Problem 1}
Page 135, problem 7.2 -- For example 2.4, you can restrict attention to the
cover $\R^2 \to T^2$. 

{\bf Problem 2} Page 144, problem 10.1  


Given a triangulated surface, we can order the vertices and describe
edges and triangles by their vertices. If a triangle has three
vertices $v_i$, $v_j$ and $v_k$, and if $i<j<k$, we denote the
triangle they span by $t_{ijk}$.  Likewise $e_{ij}$, with $i<j$, is
the edge from $v_i$ to $v_j$. Let $C_0$ be the free abelian group
whose generators are the vertices of the triangulation, let $C_1$ be
the free abelian group whose generators are the edges, and let $C_2$
be the free abelian group whose generators are the triangles. In other
words, a 2-chain is a formal sum of triangles, a 1-chain is a formal
sum of edges, and a 0-chain is a formal sum of vertices. We define
$\partial_2 t_{ijk} = e_{jk} - e_{ik} + e_{ij}$ and $\partial_1 e_{ij}
= v_j - v_i$ and extend by linearity to all chains. Let $C_k$ be zero
for $k<0$ or $k>2$, and let $\partial_k: C_k \to C_{k-1}$ be the 
zero map when $k$ is anything other than 1 or 2.  It's easy to
check that $\partial_k \circ \partial_{k+1} = 0$ for all $k$, 
so the kernel of $\partial_k$ contains the image of $\partial_{k+1}$,
and we can define 
the {\it $k$th simplicial homology} of the triangulated surface to
be the kernel of $\partial_k$ modulo the image of $\partial_{k+1}$. 
 
{\bf Problem 3} Revisit problems 7.2, 7.3, and 7.4 on page 19, only now
compute the simplicial homology of each triangulation. 

We now generalize to higher dimensions.  Working in $\R^k$, the {\it
standard $k$-simplex} is the convex hull of the points
$v_0=(0,0,\ldots,0)$, $v_1=(1,0,0,\ldots,0)$, $v_2=(0,1,0,\ldots,0)$,
through $v_k=(0,0,\ldots,0,1)$. We will label this is $\Delta^k_0$.
Now let $A$ be a rank-$k$ affine map from $\R^k$ to $\R^n$ that takes
$v_0,\ldots,v_k$ to $k+1$ distinct points $w_0,\ldots,w_k$. The image
of $\Delta^k_0$ under this map is called a $k$-simplex with vertices
$w_0,\ldots,w_k$ and is denoted $\Delta_{w_0,\ldots,w_k}$.  We order
the vertices in any simplicial complex, and generally list the
vertices of each simplex in increasing order.

(Actually, it's sometimes useful to write the vertices in arbitrary
order, with the understanding that $t_{jik}$ means $-t_{ijk}$, etc. That is, 
permuting the order of the vertices means multiplying by the
sign of the permutation.  For instance, 
$\partial t_{ijk} = e_{jk} - e_{ik} + e_{ij}$, regardless of the ordering
of $i$, $j$, and $k$. Still, we usually take
as a basis for $C^k$ the $k$-simplices with vertices written in
increasing order.)

The boundary of a simplex $\Delta_{w_0,\ldots,w_k}$, is defined to be
$\sum_{i=0}^k (-1)^i \Delta_{w_0,\ldots,w_{i-1},w_{i+1},\ldots w_k}$.  The $k-1$
simplex based on all the vertices except $w_i$ is called the $i$-th {\it face}
of the $k$-simplex.

{\bf Problem 4}. Show that $\partial_{k-1} \circ \partial_{k} = 0$. Also
show that the definition of $\partial \Delta_{w_0,\ldots,w_k}$ works even
if the vertices are NOT listed in increasing order. 

A {\it simplicial complex} is a collection of
simplices such that (1) every collection of vertices determines at most
one simplex (for instance, there can't be two edges from $v_i$ to
$v_j$), and (2) the intersection of any two simplices is the simplex
defined by their common vertices (or is empty if they have no vertices
in common), and (3) if a simplex is in the collection, then so are all
its faces, and the faces of the faces, etc. 
(e.g., if $t_{ijk}$ is a 2-simplex in the collection, then $e_{ij}$,
$e_{ik}$ and $e_{jk}$ are all 1-simplices in the collection, and
$v_i$, $v_j$ and $v_k$ are all 0-simplices.) The simplest simplicial
complex of dimension $k$ consists of a single $k$-simplex, together
with its faces and sub-faces. 

{\bf Problem 5.}  Compute the homology of the 2-simplex 
$\Delta^2_0$ (together with
its faces and sub-faces, of course). 
Repeat for $\Delta^3_0$.  Repeat for the complex obtained from
the faces of $\Delta^3_0$ without the 3-simplex itself (i.e., the surface of 
a tetrahedron). 

%{\bf Problem 6.} See if you can generalize, finding (and proving!) a formula
%for the homology of $\Delta^k_0$ (including its faces and sub-faces). 
%This one isn't easy, and involves finding
%a partial inverse to the boundary map.  Do NOT spend too much time of this. 
%If you truly get stuck, give up. 

\bye 

{\bf Problem 7.} Page 163, problem 2.1

{\bf Problem 8.} Page 166, problem 3.3.

\bye
