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\begin{document}


\section*{Lecture 4}

\subsection*{Graph of Spaces}

\begin{definition}
A graph of spaces $X$ is a topological space with a
structure dictated by a graph $\Gamma$ and certain data.
\begin{verse}
1) There is a topological space $X_v$ for each vertex $v$ in $\Gamma$.
$X_v$ is called a vertex space of $X$.\\
2) There is a topological space $X_e$ for each edge e of $\Gamma$. $X_e$
or $X_e\times I$ is edge space of $X$. ($I=[-1,1]$)\\
3)For each edge $e$ attached to vertices $i(e),\tau(e)$ there are continuous
(``attaching'') maps
$\Phi_{e-}:X_e\to X_{i(e)},\: \Phi_{e+}:X_e\to X_{\tau(e)}$.
\end{verse}
From this data we can build the space $X$ with quotient topology.
$$
X=
\left.
\begin{matrix}
&\\
&\left(\bigsqcup\limits_{\substack{\mathrm{vertices}\\ v}} X\right)\sqcup
\left(\bigsqcup\limits_{\substack{\mathrm{edges}\\ e}} X_e\times I\right)\\
&
\end{matrix}
\right/
\begin{matrix}
&\\
\Phi_{e-}(a)\sim (a,-1)\:\forall a\in X_e&\\
\Phi_{e+}(a)\sim (a,+1)\:\forall a\in X_e.&
\end{matrix}
$$
\end{definition}
Example (see figure).

<figure>

\begin{theorem}
Assume all attaching maps of edge spaces in our graph of spaces
are $\pi_1$-injective ($f:A\to B$ s.t. $f_*:\pi_1A\to \pi_1B$
is a monomorphism).

Then $\pi_1X$ ``splits'' as a graph of groups in the obvious way:
use the same graph $\Gamma$, $G_v\cong\pi_1X_v$, $G_e\cong\pi_1X_e$,
$\phi_{e_{+}}=(\Phi_{e_{+}})_*:\pi_1X_e\to\pi_1X_{\tau(e)}$,
$\phi_{e_{-}}=(\Phi_{e_{-}})_*:\pi_1X_e\to\pi_1X_{i(e)}$.

{\small (We can assume all spaces have a base point, and all the maps are base
point preserving. In fact, it doesn't matter.)}

Then $\pi_1X\cong\pi_1\mathrm{associated\: graph\: of\: groups}$.
\end{theorem}
Example (see figure ``pair of pants'').

<figure>

Example (see figure ``$\mathrm{BS}(n,m)$'').

<figure>

\subsubsection*{Reminder. How to compute $\pi_1\Gamma$}
Example (see figure ``house'').

<figure>

Pick a base point $p$ in $\Gamma$.
Pick a maximal tree $T\subset\Gamma$. Then there is a one-to-one
correspondence
``generators of $\pi_1\Gamma$ $\leftrightarrow$ edges $\Gamma\setminus T$''.

For each oriented edge $e$ in $\Gamma\setminus T$ put
$\overrightarrow{p\,i(e)}e\overleftarrow{p\,\tau(e)}=\sigma_e$.
(Let $p\in T$ be a base point. For each $v\in T$ define
$\overrightarrow{p\,v}$ to be the shortest edge path from
$p$ to $v$ in $T$.)

Then, in our example,
$\pi_1(\Gamma)=\langle\sigma_a, \sigma_b,\sigma_c\rangle$.

\subsubsection*{Arbitrary Group a Fundamental Group}
Every group is a $\pi_1X$ for some topological space $X$.

\begin{theorem}
Let $G$ be a group, let $P=\langle g_1,\ldots|R_1,\ldots\rangle$ be
its presentation, let $X$ be the standard $2$-complex of $P$.
Then $G=\pi_1X$.
\end{theorem}

{\small (BTW, proof of the theorem is Van Kampen's theorem.)}

In particular, let $M$ be the multiplication table presentation.
Let $X_G$ be the standard $2$-complex of $M$.
(This is called canonical graph of spaces.)

Now, a monomorphism
$H\subset G$ corresponds to a $\pi_1$-injective embedding
$X_H\subset X_G.$

Using this we see that every graph of groups is induced (arises)
from some graph of spaces (in fact, many graphs of spaces).

Example (see figure ``non-canonical graph of spaces'').
\end{document}
