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\begin{document}
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\parbox[b]{4in}{
  \small\url{http://drorbn.net/?title=AKT-14}
  \newline\bf
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1314}{2014}:
  \newline\href{http://drorbn.net/?title=AKT-14}{MAT 1350 --- Algebraic Knot Theory}:
}
\hfill\parbox[b]{2.1in}{\begin{flushright}
  \Large\bf The Fulton-MacPherson Compactification
\end{flushright}}

\vskip -3mm\rule{\textwidth}{1pt}

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

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

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

\begin{definition} A ``$d$-manifold with corners'' is defined in the same way as a manifold, except coordinate patches look like neighborhoods of $0$ in $\bbR^d_{+k}\coloneqq \{x\in\bbR^d\colon x^i\geq 0\text{ for }i\leq k\}$ instead of merely like neighberhoods of $0$ in  $\bbR^d$ or in $\bbR^d_+\coloneqq \{x\in\bbR^d\colon x^1\geq 0\}$.
\end{definition}

\begin{theorem} $C_A(M)$ ia a manifold with corners, and
$\displaystyle \partial C_A(M) =
  \coprod_{A'\subset A,\ |A'|\geq 2}
  \left\{(p, c)\colon p\in C^o_{A/A'}(M),\ c\in \tilde{C}_{A'}(T_{p_{A'}}M)\right\}
$.
\end{theorem}

\begin{theorem}
\begin{enumerate}
\item 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 $p_B\colon C_A(M)\to C_B(M)$. In
  particular, for every $i,j\in A$ there is a ``direction map''
  $\phi_{ij}\colon C_A(\bbR^n)\to C_{\{i,j\}}(\bbR^n)\sim S^{n-1}$.
\item If $f:M\to N$ is a smooth embedding, then there's a natural
  $f_\star:C_A(M)\to C_A(N)$.
\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 $a_0\overset{D}{\longdash}a_1$.
Likewise, if $A_{0,1}\subset A$ are disjoint subsets, we write
$A_0\overset{D}{\longdash}A_1$ if $a_0\overset{D}{\longdash}a_1$ 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
$\displaystyle
  C^o_D(M) \coloneqq \{
    p\colon A\to M\colon
    p(a_0)\neq p(a_1)\text{ whenever }a_0\overset{D}{\longdash}a_1
  \}
$.
\end{definition}

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

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

\begin{definition} Write $S^n=\bbR^n\udot\{\infty\}$ and set
$\displaystyle \bar{C}_A(\bbR^n)\coloneqq
  \left\{
    c\in C_{A\udot\{\infty\}}(S^n)\colon p_\infty(c)=\infty
  \right\}
$.
\end{definition}

\begin{theorem} $\bar{C}_A(\bbR^n)$ is a compact manifold with corners and the direction maps $\phi_{ij}\colon \bar{C}_A(\bbR^n)\to S^{n-1}$ remain well-defined.
\end{theorem}

Finally, given $\gamma\colon S^1\to\bbR^3$ and disjoint finite sets $A$ and $B$, we set
\[
  C^\gamma_{A,B}\coloneqq\left\{
    (c',c)\colon c'\in C_A(S^1),\ c\in \bar{C}_{A\cup B}(\bbR^3),\ \gamma_\ast(c')=p_A(c)
  \right\}
\]
(and similarly $C^\gamma_D$ for appropriate graphs $D$). The obvious variants of the theorems remain valid.

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