\chapter{Simplicial $\Z_2$-Homology: Physical Algebra}

\begin{exercise}
Check that $\CC_n(K)$ is an abelian group.
\end{exercise}

\oneperpage

\begin{exercise}
Verify that $\partial$ is a homomorphism, and use the definition to 
compute the $\Z_2$-boundary of $\sigma_{1}+\sigma_{2}$ in Figure \ref{twocomplex}.
\end{exercise}

\oneperpage

\begin{exercise}
Explore:
\begin{enumerate}
\item
Which $2$-chains of Figure \ref{twocomplex} are cycles?
\item
Which $1$-chains of Figure \ref{twocomplex} are cycles?
\item
Which $1$-chains of Figure \ref{twocomplex} are boundaries?
\item
Which $0$-chains of Figure \ref{twocomplex} are cycles?
\item
Which $0$-chains of Figure \ref{twocomplex} are boundaries?
% Ans: even number of vertices
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
Both $\ZZ_n(K)$ and $\BB_n(K)$ are subgroups of
$\CC_n(K)$.  
Moreover, 
$$\partial\circ\partial=0,$$
in other words, $\partial_{n}\circ\partial_{n+1}=0$ for each index $n \geq 0$. 
Hence, $\BB_n(K)\subset \ZZ_n(K)$.
\end{thm}

\oneperpage

\begin{exercise}
List all the equivalence classes of $0$-cycles, $1$-cycles and
$2$-cycles in the complex in Figure \ref{twocomplex}.
\end{exercise}

\oneperpage

\begin{exercise}
List all the equivalence classes of $0$-cycles, $1$-cycles and
$2$-cycles in a triangulated 2-sphere with its standard
triangulation as the faces of a 3-simplex.
\end{exercise}

\oneperpage

\begin{thm}
If $K$ is a one-point space, $\HH_{n}(K) \cong 0$ for $n \geq 0$ and $\HH_{0}(K) \cong \Z$.
\end{thm}

\oneperpage

\begin{thm}
If $K$ is connected, then $\HH_0(K)$ is isomorphic to $\Z_2$.
If $K$ has $r$ connected components, then $\HH_0(K)$ is
isomorphic to $\Z_2^r$.
\end{thm}

\oneperpage

\begin{exercise} 
Let $K$ be a triangulation of a $3$-dimensional ball that consists of a $3$-simplex together with its faces.  
Compute $\HH_n(K)$ for each $n$.
\end{exercise}

\oneperpage

\begin{exercise} 
Let $K$ be a triangulation of a $2$-sphere that consists of the proper faces of a $3$-simplex.  
Compute $\HH_n(K)$ for each $n$.
\end{exercise}

\oneperpage

\begin{thm}
For $x$ seeing $K$, and $\sigma$ a simplex of $K$,
$$
\partial \cone_{x} (\sigma) + \cone_{x} (\partial \sigma)  = \sigma.
$$
\end{thm}

\oneperpage

\begin{cor}
For any complex $K$ and $x$ seeing $K$, the complex $x*K$ is acyclic.
\end{cor}

\oneperpage

\begin{thm}
The complex $K$ consisting of an $n$-simplex together with all its faces is acyclic.
\end{thm}

\oneperpage

\begin{exercise}
Let $f:K\to L$ be a simplicial map. Carefully write out the definition of the natural induced map from $n$-chains of $K$ to $n$-chains of $L$: $f_{\#n}:\CC_n(K)\to \CC_n(L)$ and show that it is a homomorphism. 
\end{exercise}

\oneperpage

\begin{exercise}
If the simplicial map $f:K\to L$ maps an $n$-simplex $\sigma$ to an $(n-1)$-simplex $\tau$, what is $f_{\#n}(\sigma)$?
\end{exercise}

\oneperpage

\begin{thm}
\label{boundary-of-the-image}
Let $f:K\to L$ be a simplicial map, and let $f_\#$ be the
induced map $f_\#:\CC_n(K)\to \CC_n( L)$.  Then for any chain 
$c\in \CC_n(K)$, 
$$\partial(f_\#(c))=f_\#(\partial(c)).$$
In other words, the diagram:
$$
\begin{CD}
\CC_n(K)   @>{f_\#}>>  \CC_n(L)    \\
@V{\partial}VV      @VV{\partial}V \\
\CC_{n-1}(K)   @>{f_\#}>>   \CC_{n-1}(L)
\end{CD}
$$
commutes.
%$$
%\begin{CD}
%\CC_n(K)   @>{\partial}>>   \CC_{n-1}(K)   \\
%@V{f_\#}VV      @VV{f_\#}V \\
%\CC_n( L)   @>{\partial}>>
%\CC_{n-1}( L)
%\end{CD}
%$$
%commutes.
\end{thm}

\oneperpage

\begin{thm}
Let $f:K \to L$ be a simplicial map. Then the induced homomorphism $f_*:\HH_n(K)\to \HH_n( L)$ is a well-defined
homomorphism.
\end{thm}

\oneperpage

\begin{exercise}
Let $K$ be a complex comprising the proper faces of a hexagon: six edges and six vertices $v_0,\ldots,v_5$.  Let $L$ be the complex comprising the proper faces of a triangle: three edges and three vertices $w_0,w_1,w_2$.  Let $f$ be a simplicial map that sends $v_i$ to $w_{(i\mod 3)}$.  Compute the homology groups of $K$ and $L$ and describe the simplicial map $f$ and the induced homomorphism $f_*$.
\end{exercise}

\oneperpage

\begin{exercise}
Suggest a homomorphism from $\CC_n(K)$ to $\CC_n(\sd K)$ that commutes with $\partial$.  Could its induced homomorphism on homology be an inverse for $\lambda_{*}$?
\end{exercise}

\oneperpage

\begin{thm}
The subdivision operator commutes with the boundary operator, that is,
if $c$ is a chain in $K$, then $\sdop (\partial c) = \partial \sdop (c)$.  
\end{thm}

\oneperpage

\begin{exercise}
Show that $\lambda_{\#} \circ \sdop = \idop$, the identity map on $C_{n}(K)$, 
and therefore $\lambda_{*} \circ \sdop_{*} = \idop_{*}$, the identity map on $H_{n}(K)$.
\end{exercise}

\oneperpage

\begin{exercise}
\label{sd-lambda-homotopic-id}
Show that $\sdop \circ \lambda_{\#}$ and $\idop$, the identity map on $C_{n}(\sd K)$, induce the 
same homomorphism on homology.
\end{exercise}

\oneperpage

\begin{exercise}
If $\sigma \in \sd K$ is contained in $\tau \in K$, then $\sdop \circ \lambda_{\#} (\sigma) $ and $\idop (\sigma)$ both lie inside $\tau$.
\end{exercise}

\oneperpage

\begin{thm} 
Let $K$ be a simplicial complex. Then $\HH_n(K)$ is isomorphic to $\HH_n(\sd K)$. In fact, if the simplicial map $\lambda: \sd K \to K$ is defined by taking each vertex in $\sd K$ to any vertex of the simplex in $K$ of which it is the barycenter, then 
$$\lambda_{*}: \HH_n(K) \to \HH_n(\sd K)$$ 
is an isomorphism. 
Also, the induced homomorphism of the subdivision operator 
$$\sdop_{*}: \HH_n(K) \to \HH_n(\sd K)$$
is an isomorphism and is the inverse of $\lambda_{*}$.
\end{thm}

\oneperpage

\begin{thm} 
Let $\sd^\ell K$ and $\sd^m K$ be barycentric subdivisions of $K$. 
Suppose $g: \sd^\ell K\to L$ and $h:\sd^m K \to L$ are simplicial approximations to 
a continuous function $f:|K|\to |L|$.
Then $g_* \circ \sdop_{*}^{\ell}:\HH_n(K)\to \HH_n( L)$ is the
same homomorphism as $h_* \circ \sdop_{*}^{m}$.
\end{thm}

\oneperpage

\begin{lem} If $K$, $L$, and $M$ are simplicial complexes and $f:|K|\to |L|$ and
$g:|L|\to |M|$ are continuous maps, then $(g\circ f)_*=g_*\circ f_*$.
\end{lem}

\oneperpage

\begin{lem} If $i:|K|\to |K|$ is the identity map, then $i_*$ is the identity homomorphism on each homology group.\end{lem}

\oneperpage

\begin{thm}
Let $K$ and $L$ be simplicial complexes. If $f:|K|\to |L|$ is a homeomorphism, 
then $f$ induces an isomorphism between the $\Z_2$-homology groups of $K$ and $L$.
\end{thm}

\oneperpage

\begin{thm}
Let $K$ and $L$ be simplicial complexes. If $f:|K|\to |L|$ is a homotopy equivalence, 
then $f$ induces an isomorphism between the $\Z_2$-homology groups of $K$ and $L$.
\end{thm}

\oneperpage

\begin{cor}
\label{sdr}
If $K$ is a strong deformation retract of $L$. Then $K$ and $L$ have
isomorphic $\Z_2$-homologies.
\end{cor}

\oneperpage

\begin{exercise}
If $K$ is a finite simplicial complex, verify that the intersection of two subcomplexes of $K$ is a subcomplex.
\end{exercise}

\oneperpage

\begin{exercise}
Note that a cycle in $A \cap B$ is still a cycle in $A$, $B$, and $K$.  Then answer:
\begin{enumerate}
\item Can a trivial cycle in $A \cap B$ be non-trivial in $A$?  
\item Can a non-trivial cycle in $A \cap B$ be trivial in $A$?  
\item Can a non-trivial cycle in $A \cap B$ that's also non-trivial in $A$ and in $B$ be trivial in $K$?
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
Let $K$ be a finite simplicial complex and $A$ and $B$ be
subcomplexes such that  $K=A\cup B$. 
If $\alpha$, $\beta$ are $k$-cycles in $A$ and $B$ respectively,
and if $\alpha \sim_{\Z_2} \beta$ in $K$, 
then there is a $k$-cycle $c$ in $A\cap B$ 
such that $\alpha \sim_{\Z_2} c$ in $A$ and
$\beta \sim_{\Z_2} c$ in $B$.
\end{thm}

\oneperpage

\begin{thm}
Let $K$ be a finite simplicial complex and $A$ and $B$ be
subcomplexes such that  $K=A\cup B$. Let $z$ be a $k$-cycle in $K$. Then there exist
$k$-chains $\alpha$ and $\beta$
in $A$ and $B$ respectively such that: \be
\item[1.] $z=\alpha+\beta$ and
\item[2.] $\partial \alpha=\partial \beta$ is a $(n-1)$-cycle $c$ in $A\cap B$. \ee

Furthermore, if $z= \alpha'+\beta'$, a sum of $n$-chains in $A$ and $B$ respectively, 
and $c'= \partial \alpha' = \partial \beta'$ is a 
$(n-1)$-cycle, then $c'$ is homologous to $c$ in $A \cap B$.
\end{thm}

\oneperpage

\begin{exercise}
Let $K$ be a simplicial complex and $A$ and $B$ be subcomplexes
such that  $K=A\cup B$. Construct natural homomorphisms $\phi,\psi,\delta$ among the groups below and show that 
$\psi \circ \phi =0$ and $\delta \circ \psi = 0$. 
\begin{enumerate}
\item $\phi: \HH_n(A\cap B)\to \HH_n(A)\oplus \HH_n(B)$.
\item $\psi: \HH_n(A)\oplus \HH_n(B)\to \HH_n(K)$.
\item $\delta: \HH_n(K)\to \HH_{n-1}(A\cap B)$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}[$\Z_2$ Mayer-Vietoris]
\index{Mayer-Vietoris Theorem!$\Z_2$} Let $K$ be a finite simplicial
complex and $A$ and $B$ be subcomplexes such that  $K=A\cup
B$.  The sequence 
$$
\dots \to \HH_n(A\cap B)  \to \HH_n(A)\oplus \HH_n(B) \to \HH_n(K) \to \HH_{n-1}(A\cap B) \to \dots
$$ 
using the homomorphisms $\phi,\psi,\delta$ above, is exact.
\end{thm}

\oneperpage

\begin{exercise}
Let $C, D, E$ be groups, and arrows represent homomorphisms below.
\begin{enumerate}
\item $0\to C\stackrel{\phi}{\to}D$ is exact at $C$ if and only if $\phi$ is one-to-one.
\item $D\stackrel{\psi}{\to}E\to 0$ is exact at $E$ if and only if
$\psi$ is onto.
\item $0\to C\stackrel{\phi}{\to}D\to 0$ is exact if and only if
$\phi$ is an isomorphism.
% The following is not true!
%\item $0\to C\stackrel{\phi}{\to}D \stackrel{\psi}{\to}E\to0$ is
%exact if and only if $\psi$ induces an isomorphism $E\cong D/\phi(C)$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
Compute the $\Z_2$-homology groups
%(of both types)
for each complex $K$
below.
\begin{enumerate}
\item The bouquet of $k$ circles (the union of $k$ circles identified at a point).
\item A wedge of a $2$-sphere and a circle (the two spaces are glued at one point).
\item A $2$-sphere union its equatorial disk.
\item A double solid torus.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
Compute the $\Z_2$-homology groups of a torus using Mayer-Vietoris in two different ways (with two different decompositions).
\end{exercise}

\oneperpage

\begin{exercise}
Use the Mayer-Vietoris Theorem to compute $\HH_n(M)$ 
for every compact, triangulated $2$-manifold $M$.
What compact, triangulated $2$-manifolds are not distinguished from
one another by $\Z_2$-homology?  What does $\HH_2(M)$ tell you?
\end{exercise}

\oneperpage

\begin{exercise} 
Let $p,q\in\Z$ be relatively prime.  Calculate $\HH_n(L(p,q))$, the homology of the lens space $L(p,q)$.
\end{exercise}

\oneperpage

\begin{exercise}
Use the Mayer-Vietoris Theorem to compute $\HH_n(K)$ 
for the complexes $K$ pictured in Figure \ref{twospaces}.

\begin{figure}[h]
\begin{center}
%\includegraphics[width = 5in]{Figures/twospaces}
\caption{\label{twospaces} Two interesting spaces.}
\end{center}
\end{figure}

\end{exercise}

\oneperpage

\begin{exercise}
Use the Mayer-Vietoris Theorem to find the $Z_2$-homology groups for each of the following spaces.
\begin{enumerate}
\item $\Sph^n$.
\item A cone over a finite simplicial complex $K$.
\item A suspension over a finite simplicial complex $K$
(that is, the finite simplicial complex created by gluing two cones over $K$ along $K$).
\item[4.] $\PP^n$ (which is $\Sph^n$ with antipodal points identified).
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
Let $K$ be a simplicial complex where $T=\{\sigma_i\}_{i=1}^k$.
Then $$K=\bigsqcup_{i=1}^k\interior(\sigma_i),$$ where $\sqcup$
denotes disjoint union.
\end{thm}

\oneperpage

\begin{exercise}
Let $K$ be a $3$-simplex with triangulation
shown (a tetrahedron).  
Find an open cell decomposition of $K$
with one vertex, one open $2$-cell, and one
open $3$-cell. This example
shows that it is not necessary to have every dimension less then the
dimension of $K$ represented.
\end{exercise}

\oneperpage

\begin{thm}
Both $\ZZ_n^c(K^c)$ and $\BB_n^c(K^c)$ are subgroups of
$\CC_n^c(K^c)$.  Moreover, $\BB_n^c(K^c)\subset
\ZZ_n^c(K^c)$.
\end{thm}

\oneperpage

\begin{thm}
Let $K^c$ be an open cell decomposition of the finite simplicial complex $K$. 
Then for each $n$, the obvious homomorphism $\HH_n^c(K^c)\to \HH_n(K)$ is an isomorphism.
\end{thm}

\oneperpage

\begin{exercise}
For each space below, describe a triangulation $K$ and an open cell decomposition $K^c$.
Then use cellular homology to compute $\HH_n(K)$ for each $n$:
\begin{enumerate}
\item The sphere.
\item The torus.
\item The projective plane.
\item The Klein bottle.
\item The double torus.
\item Any compact, connected, triangulated $2$-manifold.
\item The M\"{o}bius band.
\item The annulus.
\item Two (hollow) triangles joined at a vertex.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
What is $\HH_n(\Sph^k)$ for $n=0,1,2,\ldots$ and
$k=0,1,2,\ldots$?
\end{exercise}

\oneperpage

\begin{exercise}
What is $\HH_n(\T)$ for $n=0,1,2,\ldots$ for a solid torus $\T$?
\end{exercise}

\oneperpage

