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\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}

\def\bbR{{\mathbb R}}
\newcommand{\udot}{{\mathaccent\cdot\cup}}

\begin{document}

\title{The Fulton-MacPherson Compactification}
\author{Dror Bar-Natan}
\date{\today}
\thanks{This handout is at available at \url{http://drorbn.net/?title=AKT-14}.}

\maketitle

Let $M$ be a manifold and let $A$ be a finite set.

\begin{definition} The open configuration space of $A$ in $M$ is
\[ C^o_A(M) := \{\text{injections }\iota:A\to M\}. \]
\end{definition}

\begin{definition} The compactified configuration space of $A$ in $M$ is
\[ C_A(M) :=
  \coprod_{\substack{
    \{A_1,\dots,A_k\} \\
    A=\udot A_\alpha
  }}
  \left\{
    \left(
      p_\alpha\in M, c_\alpha\in \tilde{C}_{A_\alpha}(T_{p_\alpha}M)
    \right)_{\alpha=1}^k:
    p_\alpha\neq p_\beta\text{ for }\alpha\neq\beta
  \right\}
\]
where if $V$ is a vector space and $|A|\geq 2$,
\[ \tilde{C}_A(V) := \coprod_{\substack{
    \{A_1,\dots,A_k\} \\
    A=\udot A_\alpha;\ k\geq 2
  }}
  \left\{
    \left(
      v_\alpha\in V, c_\alpha\in \tilde{C}_{A_\alpha}(T_{v_\alpha}V)
    \right)_{\alpha=1}^k:
    v_\alpha\neq v_\beta\text{ for }\alpha\neq\beta
  \right\}
  \left/\parbox{1in}{\centering translations and dilations.}\right.
\]
while if $A$ is a singleton, $\tilde{C}_A(V):=\{\text{a point}\}$.
\end{definition}

\begin{theorem}
\begin{enumerate}
\item $C_A(M)$ ia a manifold with corners, and if $M$ is compact, so is
  $C_A(M)$.
\item If $A$ is a singleton, $C_A(M)=M$. If $A$ is a doubleton, then
  $C_A(M)$ is isomorphic to $M\times M$ minus a tubular neighborhood
  of the diagonal $\Delta\subset M\times M$. That is, $C_A(M)=M\times
  M-V(\Delta)$.
\item If $B\subset A$ then there is a natural map $C_A(M)\to C_B(M)$. In
  particular, for every $i,j\in A$ there is a map
  $\phi_{ij}:C_A(\bbR^3)\to C_{\{i,j\}}(\bbR^3)\sim S^2$.
\item If $f:M\to N$ is a smooth embedding, then there's a natural
  $f_\star:C_A(M)\to C_A(N)$. \qed
\end{enumerate}
\end{theorem}

Now let $D$ be a graph whose set of vertices
is $A$.  If two different vertices $a_{0,1}\in A$ are connected
by an edge in $D$, we write $a_0\overset{D}{-}a_1$.
Likewise, if $A_{0,1}\subset A$ are disjoint subsets, we write
$A_0\overset{D}{-}A_1$ if $a_0\overset{D}{-}a_1$ for some
$a_0\in A_0$ and $a_1\in A_1$.  For any subset $A_0$ of $A$ we let
$D(A_0)$ be the restriction of $D$ to $A_0$.

\begin{definition} The open configuration space of $D$ in $M$ is
\[
  C^o_D(M) := \{
    \iota:A\to M:\,
    \iota(a_0)\neq\iota(a_1)\text{ whenever }a_0\overset{D}{-}a_1
  \}.
\]
\end{definition}

\begin{definition} The compactified configuration space of $D$ in $M$ is
\[ C_D(M) :=
  \coprod_{\substack{
    \{A_1,\dots,A_k\} \\
    A=\udot A_\alpha \\
    \forall\alpha\ D(A_\alpha)\text{ connected}
  }}
  \left\{
    \left(
      p_\alpha\in M, c_\alpha\in \tilde{C}_{D(A_\alpha)}(T_{p_\alpha}M)
    \right)_{\alpha=1}^k:
    p_\alpha\neq p_\beta\text{ whenever }A_\alpha\overset{D}{-}A_\beta
  \right\}
\]
where if $V$ is a vector space and $|A|\geq 2$,
\[ \def\neg{\hspace{-2mm}}
  \tilde{C}_D(V) := \neg \coprod_{\substack{
    \{A_1,\dots,A_k\} \\
    A=\udot A_\alpha;\ k\geq 2 \\
    \forall\alpha\ D(A_\alpha)\text{ connected}
  }} \neg
  \left\{
    \left(
      v_\alpha\in V, c_\alpha\in \tilde{C}_{D(A_\alpha)}(T_{v_\alpha}V)
    \right)_{\alpha=1}^k:
    v_\alpha\neq v_\beta\text{ whenever }A_\alpha\overset{D}{-}A_\beta
  \right\}
  \left/\parbox{0.8in}{\centering translations and dilations.}\right.
\]
while if $A$ is a singleton, $\tilde{C}_D(V):=\{\text{a point}\}$.
\end{definition}

\begin{theorem} The obvious parallel of the previous theorem holds. \qed
\end{theorem}

\end{document}