\chapter{Singular Homology: Abstracting Objects to Maps}

\begin{thm}[Eilenberg-Steenrod]
Fix a group $G$.  Any two homology theories on compact triangulable pairs with coefficient group $G$ are isomorphic.
\end{thm}

\oneperpage

\begin{exercise}
Let $X \subset \E$ be star-convex with respect to a point $x$.  
Verify that $\Phi^{n+1}_{0} \circ \coneop_{x}$ is the identity map on $S_{n}(X)$.
\end{exercise}

\oneperpage

\begin{thm}
For all $n\geq 0$, 
$$\partial_{n}\circ\partial_{n+1}=0.$$
%$$\partial \circ\partial =0.$$
Hence $\image \partial_{n+1} \subset \kernel \partial_{n}$.
\end{thm}

\oneperpage

\begin{thm}[Dimension Axiom]
\label{axiom:dimension}
If $P$ is a point, $H_{n}(P) \cong 0$ for all $n > 0$, and $H_{0}(P) \cong \Z$.
\end{thm}

\oneperpage

\begin{thm}
\label{cone-homotopy}
Let $X \subset \E$ be star convex with respect to $x \in X$.  For any singular $n$-simplex $\sigma$, 
$$
\partial_{n+1} (\coneop_{x} \sigma) + \coneop_{x} (\partial_{n} \sigma) = \sigma.
$$
\end{thm}

\oneperpage

\begin{thm}
\label{star-acyclic}
Show that any star-convex space is acyclic.
\end{thm}

\oneperpage

\begin{thm}
For a space $X$, show that $H_{0}(X)$ is a free abelian group with a generator for every path-connected component of $X$.
\end{thm}

\oneperpage

\begin{thm}
Let $f:X \to Y$ be a continuous map.  
Then for any chain $c\in S_n(X)$, $\partial(f_\#(c))=f_\#(\partial(c))$. In other words,
the diagram:
$$
\begin{CD}
S_n(X)   @>{f_\#}>>  S_n(Y)    \\
@V{\partial}VV      @VV{\partial}V \\
S_{n-1}(X)   @>{f_\#}>>   S_{n-1}(Y)
\end{CD}
$$
commutes.
\end{thm}

\oneperpage

\begin{exercise}
Check that the induced homomorphism is well-defined and a homomorphism.
\end{exercise}

\oneperpage

\begin{thm}
The identity map $i:X \to X$ induces the identity homomorphism on each homology group.
\end{thm}

\oneperpage

\begin{thm}
If $f:X \to Y$ and $g:Y \to Z$ are continuous maps between topological spaces, then $(g\circ f)_*=g_*\circ f_*$
\end{thm}

\oneperpage

\begin{thm}
If $f: X \to Y$ is a homeomorphism, then $f_{*}: H_{n}(X) \to H_{n}(Y)$ is an isomorphism between singular homology groups.
\end{thm}

\oneperpage

\begin{thm}[Homotopy Axiom]
If $f$ and $g$ are homotopic maps from $X$ to $Y$, then they induce the same homomorphism in homology.
\end{thm}

\oneperpage

\begin{thm}
\label{relative-boundary}
There is a boundary map 
$$\partial: S_{n}(X,A) \to S_{n-1}(X,A)$$
such that $\partial_{n} \circ \partial_{n+1} = 0$ for all $n \geq 0$.
\end{thm}

\oneperpage

\begin{exercise}
Check that if $A= \emptyset$, the empty set, then $H_{n}(X,A)= H_{n}(X)$, the usual homology.
\end{exercise}

\oneperpage

\begin{exercise}
Let $X$ be the complex consisting of a triangle and all its faces.  Determine $H_{n}(X,A)$ for all $n \geq 0$.
\end{exercise}

\oneperpage

\begin{exercise}
Let $K$ be a triangulation of an annulus, and let $K'$ be the subcomplex consisting of the inner and outer edges of the annulus.  Find a relative $1$-cycle in $C_{1}(X,A)$  that is not a relative $1$-boundary.
\end{exercise}

\oneperpage

\begin{exercise}
Let $K$ be a triangulation of a M\"obius band, and let $K'$ be its boundary.  Determine $H_{n}(X,A)$ for $n\geq 0$.
\end{exercise}

\oneperpage

\begin{thm}
Given a continuous map $f:(X,A) \to (Y,B)$ there is 
an associated chain map $f_{\#}: S_{n}(X,A) \to S_{n}(Y,B)$ 
and induced homomorphism $f_{*}: H_{n}(X,A) \to H_{n}(Y,B)$.
\end{thm}

\oneperpage

\begin{thm}[Long Exact Sequence of a Pair]
Let $A$ be a subspace of $X$.  Then there is a long exact sequence:
$$
\begin{CD}
\cdots @>{\partial_{*}}>> H_{n}(A)   @>{i_{*}}>> H_{n}(X) @>{\pi_{*}}>> H_{n}(X,A) @>{\partial_{*}}>>  H_{n-1}(A) @>{i_{*}}>> \cdots
\end{CD}
$$
where the maps are induced by the inclusion maps $i:A \to X$ and $\pi: (X,\emptyset) \to (X,A)$ and the boundary map $\partial: S_{n}(X) \to S_{n-1}(X)$.
\end{thm}

\oneperpage

\begin{exercise}
Verify that $\sdop$ is a chain map, commuting with $\partial$, and verify that it is natural, which means that for any continuous map $f:X \to Y$, the following diagram commutes.
$$
\begin{CD}
S_n(X)   @>{f_\#}>>  S_n(Y)    \\
@V{\sdop}VV      @VV{\sdop}V \\
S_{n}(X)   @>{f_\#}>>   S_{n}(Y)
\end{CD}
$$.
\end{exercise}

\oneperpage

\begin{thm}
Let $\mathcal{U}$ be an open cover of a space $X$.  For any singular simplex $\sigma$, there exists an $m$ such that each term of $\sdop^{m}(\sigma)$ has an image that lies within one of the elements of $\mathcal{U}$.
\end{thm}

\oneperpage

\begin{thm}
There is a chain homotopy between $\sdop^{m}$ and the identity map on $S_{n}(X)$; in other words, there exists a homomorphism $D_{X}:S_{n}(X) \to S_{n+1}(X)$ such that
$$
\partial D_{X} \sigma + D_{X} \partial \sigma = \sdop^{m} \sigma - \sigma
$$
for every singular simplex $\sigma$ of $X$.  Moreover, this chain homotopy is natural, meaning it commutes with maps of spaces: if $f:X \to Y$, then $f_{\#} \circ D_{X} = D_{Y} \circ f_{\#}$.
\end{thm}

\oneperpage

\begin{thm}
For each $n \geq 0$, the induced homomorphism
$$
\sdop^{m}_{*}:H_{n}(X) \to H_{n}(X)
$$
is an isomorphism.
\end{thm}

\oneperpage

\begin{thm}
Suppose $U$ and $A$ are subspaces of $X$ such that $\overline{U} \subset \interior A$.  Then the inclusion map of $(X-U,A-U)$ in $(X,A)$ induces an isomorphism
$$
H_{n}(X-U,A-U) \cong H_{n}(X,A).
$$
\end{thm}

\oneperpage

\begin{thm}
For the $n \geq 1$, 
show that $H_{n}(S^{n}) \cong \Z$ and $H_{n}(S^{k}) \cong 0$ if $n \neq k$.
\end{thm}

\oneperpage

