
\begin{itemize}
\item $C_k(M)$:  $k$-chains.
\item $Z_k(M)$:  $k$-cycles ($\partial c = 0$).
\item $B_k(M)$:  $k$-boundaries ($c = \partial b$).
\item $H_k(M)$:  $Z_k(M) / B_k(M)$ ($k$th homology is $k$-cycles mod
$k$-boundaries).
\end{itemize}

\begin{itemize}
\item $\Omega^k(M)$:  $k$-forms.
\item $Z^k(M)$:  closed $k$-forms ($d\omega = 0$).
\item $B^k(M)$:  exact $k$-forms ($\omega = d\eta$).
\item $H^k(M)$:  $Z^k(M) / B^k(M)$.  ($k$th cohomology is closed $k$ forms
	mod exact $k$-forms).
\end{itemize}

Containments and quotients (since $\partial^2=0$ and $d^2=0$):
\begin{eqnarray*}
	&&C_k(M) \supseteq Z_k(M) \supseteq B_k(M); \qquad H_k(M) = Z_k(M) / B_k(M) \\
	&&\Omega^k(M) \supseteq Z^k(M) \supseteq B^k(M); \qquad H^k(M) = Z^k(M) / B^k(M)
\end{eqnarray*}

%% ----------------------------------------------------------------
Inclusions and their induced maps on chains and homology (used, with varying
connectivity hypotheses, in Seifert-van Kampen and Mayer-Vietoris):

\setlength{\unitlength}{1.0in}
\begin{picture}(6,1.0)
%\put(0,0){\framebox(6,1)}

\put(0.70, 0.90){$M$}
\put(0.20, 0.50){$U$}
\put(1.20, 0.50){$V$}
\put(0.60, 0.10){$U\cap V$}
\put(0.35, 0.60){\vector( 1, 1){.30}}
\put(1.15, 0.60){\vector(-1, 1){.30}}
\put(0.65, 0.22){\vector(-1, 1){.30}}
\put(0.85, 0.22){\vector( 1, 1){.30}}
\put(0.25, 0.75){$j_U$}
\put(1.15, 0.75){$j_V$}
\put(0.25, 0.25){$i_U$}
\put(1.15, 0.25){$i_V$}

\put(2.60, 0.90){$C_k(M)$}
\put(1.90, 0.50){$C_k(U)$}
\put(3.20, 0.50){$C_k(V)$}
\put(2.40, 0.10){$C_k(U\cap V)$}
\put(2.35, 0.60){\vector( 1, 1){.25}}
\put(3.15, 0.60){\vector(-1, 1){.25}}
\put(2.65, 0.22){\vector(-1, 1){.30}}
\put(2.85, 0.22){\vector( 1, 1){.30}}
\put(2.15, 0.75){$j_{U\#}$}
\put(3.15, 0.75){$j_{V\#}$}
\put(2.15, 0.25){$i_{U\#}$}
\put(3.15, 0.25){$i_{V\#}$}

\put(4.60, 0.90){$H_k(M)$}
\put(3.90, 0.50){$H_k(U)$}
\put(5.20, 0.50){$H_k(V)$}
\put(4.40, 0.10){$H_k(U\cap V)$}
\put(4.35, 0.60){\vector( 1, 1){.25}}
\put(5.15, 0.60){\vector(-1, 1){.25}}
\put(4.65, 0.22){\vector(-1, 1){.30}}
\put(4.85, 0.22){\vector( 1, 1){.30}}
\put(4.15, 0.75){$j_{U*}$}
\put(5.15, 0.75){$j_{V*}$}
\put(4.15, 0.25){$i_{U*}$}
\put(5.15, 0.25){$i_{V*}$}

\end{picture}

%% ----------------------------------------------------------------
Inclusions and their induced maps on forms and cohomology:

\setlength{\unitlength}{1.0in}
\begin{picture}(6,1.0)
%\put(0,0){\framebox(6,1)}

\put(0.70, 0.90){$M$}
\put(0.20, 0.50){$U$}
\put(1.20, 0.50){$V$}
\put(0.60, 0.10){$U\cap V$}
\put(0.35, 0.60){\vector( 1, 1){.30}}
\put(1.15, 0.60){\vector(-1, 1){.30}}
\put(0.65, 0.22){\vector(-1, 1){.30}}
\put(0.85, 0.22){\vector( 1, 1){.30}}
\put(0.25, 0.75){$j_U$}
\put(1.15, 0.75){$j_V$}
\put(0.25, 0.25){$i_U$}
\put(1.15, 0.25){$i_V$}

\put(2.60, 0.90){$\Omega^k(M)$}
\put(1.90, 0.50){$\Omega^k(U)$}
\put(3.20, 0.50){$\Omega^k(V)$}
\put(2.40, 0.10){$\Omega^k(U\cap V)$}
\put(2.65, 0.85){\vector(-1,-1){.30}}
\put(2.85, 0.85){\vector( 1,-1){.30}}
\put(2.35, 0.52){\vector( 1,-1){.30}}
\put(3.15, 0.52){\vector(-1,-1){.30}}
\put(2.15, 0.75){$j_{U}^{\#}$}
\put(3.15, 0.75){$j_{V}^{\#}$}
\put(2.15, 0.25){$i_{U}^{\#}$}
\put(3.15, 0.25){$i_{V}^{\#}$}

\put(4.60, 0.90){$H^k(M)$}
\put(3.90, 0.50){$H^k(U)$}
\put(5.20, 0.50){$H^k(V)$}
\put(4.40, 0.10){$H^k(U\cap V)$}
\put(4.65, 0.85){\vector(-1,-1){.30}}
\put(4.85, 0.85){\vector( 1,-1){.30}}
\put(4.35, 0.52){\vector( 1,-1){.30}}
\put(5.15, 0.52){\vector(-1,-1){.30}}
\put(4.15, 0.75){$j_{U}^{*}$}
\put(5.15, 0.75){$j_{V}^{*}$}
\put(4.15, 0.25){$i_{U}^{*}$}
\put(5.15, 0.25){$i_{V}^{*}$}
\end{picture}

Behavior of induced maps:
\begin{itemize}
\item Induced maps on chains are \emph{inclusions}.  E.g. a chain $c$
	in $U \cap V$ is a chain in $U$ by inclusion: $i_{U\#}(c) = c$.
\item Induced maps on forms are \emph{restrictions}.  E.g. a form $\omega$
	on $M$ is a form on $U$ by restriction: $j_{U}^{\#}(\omega) = \omega|_U$.
\end{itemize}

(See next page.)

%% ----------------------------------------------------------------
\newpage

Short exact sequences of chain maps:
$$
\begin{matrix}
	&
	&\vdots&
	&\vdots&
	&\vdots&
		\\
	0 &\longto &
	C_k(U\cap V)
		&\stackrel{i_{U\#} \oplus i_{V\#}}{\longto}&
	C_k(U) \oplus C_k(V)
		&\stackrel{j_{U\#} - j_{V\#}}{\longto}&
	C_k(M) &\longto & 0
		\\
	&
	&\downarrow \partial&
	&\downarrow \partial&
	&\downarrow \partial&
		\\
	0 &\longto &
	C_{k-1}(U\cap V)
		&\stackrel{i_{U\#} \oplus i_{V\#}}{\longto}&
	C_{k-1}(U) \oplus C_{k-1}(V)
		&\stackrel{j_{U\#} - j_{V\#}}{\longto}&
	C_{k-1}(M) &\longto& 0
		\\
	&
	&\vdots&
	&\vdots&
	&\vdots&
		\\
\end{matrix}
$$

Long exact sequence of homology:
$$
\begin{matrix}
	&\vdots&
	&\vdots&
	&\vdots&
		\\
	&H_k(U\cap V)
		&\stackrel{i_{U*} \oplus i_{V*}}{\longto}&
	H_k(U) \oplus H_k(V)
		&\stackrel{j_{U*} - j_{V*}}{\longto}&
	H_k(M) && \to\partial_*&\searrow
		\\
	&&&&&&&& \downarrow
		\\
	\swarrow&\from&\from&\from&\from&\from&\from&\from&\swarrow
		\\
	\downarrow&&&&&&&&
		\\
	\searrow&H_{k-1}(U\cap V)
		&\stackrel{i_{U*} \oplus i_{V*}}{\longto}&
	H_{k-1}(U) \oplus H_{k-1}(V)
		&\stackrel{j_{U*} - j_{V*}}{\longto}&
	H_{k-1}(M)
		\\
	&\vdots&
	&\vdots&
	&\vdots&
		\\
\end{matrix}
$$

%% ----------------------------------------------------------------
Short exact sequences of form maps:
$$
\begin{matrix}
	&
	&\vdots&
	&\vdots&
	&\vdots&
		\\
	0 &\longfrom &
	\Omega^k(U\cap V)
		&\stackrel{i_{U}^{\#} - i_{V}^{\#}}{\longleftarrow}&
	\Omega^k(U) \oplus \Omega^k(V)
		&\stackrel{j_{U}^{\#} \oplus j_{V}^{\#}}{\longfrom}&
	\Omega^k(M) &\longfrom & 0
		\\
	&
	&\uparrow d&
	&\uparrow d&
	&\uparrow d&
		\\
	0 &\longfrom &
	\Omega^{k-1}(U\cap V)
		&\stackrel{i_{U}^{\#} - i_{V}^{\#}}{\longfrom}&
	\Omega^{k-1}(U) \oplus \Omega^{k-1}(V)
		&\stackrel{j_{U}^{\#} \oplus j_{V}^{\#}}{\longfrom}&
	\Omega^{k-1}(M) &\longfrom& 0
		\\
	&
	&\vdots&
	&\vdots&
	&\vdots&
		\\
\end{matrix}
$$

\newpage

%asdfadfadfdfyy
%
%\xymatrix{
%	{\circ}
%		\ar `r[d]  ^a
%		`[rr]      ^b
%		`/4pt[rr]  ^c
%		`[rrr]     ^d
%		`_dl[drrr] ^e
%		[drrr]     ^f
%	& {\circ} & {\circ} & {\circ} \\
%	{\circ} & {\circ} & {\circ} & {\circ}
%}

%\xymatrix{
%	\vdots
%		& &
%	\vdots
%		& &
%	\vdots
%		&
%		\\
%%
%	H^k(U\cap V)
%		&\stackrel{i_{U}^{*} - i_{V}^{*}}{\longleftarrow}&
%	H^k(U) \oplus H^k(V)
%		&\stackrel{j_{U}^{*} \oplus j_{V}^{*}}{\longfrom}&
%	H^k(M)
%		&
%	\delta
%		\\
%%
%	H^{k-1}(U\cap V)
%		&\stackrel{i_{U}^{*} - i_{V}^{*}}{\longfrom}&
%	H^{k-1}(U) \oplus H^{k-1}(V)
%		&\stackrel{j_{U}^{*} \oplus j_{V}^{*}}{\longfrom}&
%	H^{k-1}(M)
%		\\
%%
%	\vdots
%		& &
%	\vdots
%		& &
%	\vdots
%		&
%		\\
%}

Long exact sequence of cohomology:
$$
\begin{matrix}
	&\vdots&
	&\vdots&
	&\vdots&
		\\
	&H^k(U\cap V)
		&\stackrel{i_{U}^{*} - i_{V}^{*}}{\longleftarrow}&
	H^k(U) \oplus H^k(V)
		&\stackrel{j_{U}^{*} \oplus j_{V}^{*}}{\longfrom}&
	H^k(M) && \from\delta & \nwarrow
		\\
	&&&&&&&& \uparrow
		\\
	\nearrow&\to&\to&\to&\to&\to&\to&\to&\nearrow
		\\
	\uparrow&&&&&&&&
		\\
	\nwarrow&H^{k-1}(U\cap V)
		&\stackrel{i_{U}^{*} - i_{V}^{*}}{\longfrom}&
	H^{k-1}(U) \oplus H^{k-1}(V)
		&\stackrel{j_{U}^{*} \oplus j_{V}^{*}}{\longfrom}&
	H^{k-1}(M)
		\\
	&\vdots&
	&\vdots&
	&\vdots&
		\\
\end{matrix}
$$

asdfadfadfdfyy

\xymatrix{
	{\circ}
		\ar
		`r[d]      ^a
		`[rr]      ^b
		`[rr]      ^c
		`[rrr]     ^d
		`_dl[drrr] ^e
		[drrr]     ^f
	& {\circ} & {\circ} & {\circ} \\
	{\circ} & {\circ} & {\circ} & {\circ}
}

asdfasdfdfdfasdf

\xymatrix{
	{\circ}
		\ar
		[rrr]      ^a
		%`[rr]      ^b
		%`[rr]      ^c
		%`[rrr]     ^d
		%`_dl[drrr] ^e
		%[drrr]     ^f
	& {\circ} & {\circ} & {\circ} \\
	{\circ} & {\circ} & {\circ} & {\circ}
}


Long exact sequence of cohomology:

\xymatrix{
	&\vdots
		& &
	\vdots
		& &
	\vdots
		& \\
%
	&H^k(U\cap V)
		&\stackrel{i_{U}^{*} - i_{V}^{*}}{\longleftarrow}&
	H^k(U) \oplus H^k(V)
		&\stackrel{j_{U}^{*} \oplus j_{V}^{*}}{\longfrom}&
	H^k(M)
		&\\
%
	% Long arrow starts in this cell
	\ar
		`l[u]
		`_r
		^\delta
		`^u[rrrrrr]
		`^l[urrrrr]
		[lurrrrrr]
%
	&H^{k-1}(U\cap V)
		&\stackrel{i_{U}^{*} - i_{V}^{*}}{\longfrom}&
	H^{k-1}(U) \oplus H^{k-1}(V)
		&\stackrel{j_{U}^{*} \oplus j_{V}^{*}}{\longfrom}&
	H^{k-1}(M)
		&\\
%
	&\vdots
		& &
	\vdots
		& &
	\vdots
		&\\
}

