## Friday, August 9, 2013

### The Fulton-MacPherson compactification of a configuration space

$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\DeclareMathOperator{\Bl}{\boldsymbol{Bl}}$

Suppose that $M$ is a real analytic manifold of dimension $m$. Fix a finite set $L$ of labels. For any subset $S\subset L$ we define the following objects.

•  The manifold $M^S$ consisting of  maps $S\to M$. We will indicate a point in $M^S$ as a collection $x_S:=(x_s)_{s\in S}$,  $x_s\in M$, $\forall s\in S$.
•  The configuration space $M(S)\subset$ consisting of injective maps $S\to M$.
• The  thin diagonal  $\Delta_S\subset M^S$ consisting of the constant maps $S\to M$.

The space of configurations $M(L)$ is an open subset of $M^L$. We want to construct   a certain  completion  $M[L]$ of $M(L)$ as a    manifold with corners.  This completion is known as the  Fulton-MacPherson compactification of $M$.  The completion $M[L]$ is compact when $M$ is compact.  We follow closely the approach   of Axelrod and  Singer, Chern-Simons perturbation theory.II,  J. Diff. Geom., 39(1994), 173-213.

We begin with some simple observations. Observe  that if $S\subset S'$, then we  have a natural projection $\pi_S: M^{S'}\to M^S$ which associates to a map $S'\to S$ its restriction to $S$

$$M^{S'}\ni x_{S'}\mapsto x_{S}\in M^S.$$

$\newcommand{\hra}{\hookrightarrow}$  We set

$$\Delta_S^L=\pi_S^{-1}(\Delta_S)\subset M^L.$$

More explicitly, $\Delta_S^L$ consists of the maps $L\to M$ which are constant on $S$. Observe that

$$M^L\setminus M(L)=\bigcup_{|S|\geq 2} \Delta_S^L.$$

For $x\in M$ and $S\subset M$ we denote by $x^S$ the constant map $S\to\lbrace x\rbrace$ viewed as an element in $M^S$.

The thin diagonal  $\Delta_S$ is a submanifold in $M^S$ of codimension $m(|S|-1)$.  We denote by $\newcommand{\eN}{\mathscr{N}}$ $\eN_S$ the normal bundle of the embedding $\Delta_S\hra M^S$.    The  fiber of the normal bundle  $\eN_S$ at a point $x^S$ is the quotient of $(T_xM)^S$ modulo the equivalence relation $\newcommand{\Llra}{\Longleftrightarrow}$

$$(u_S)_{s\in S}\in (T_xM)^S\sim (v_S)_{s\in S}\in (T_xM)^S\Llra (u_{s_0}-u_{s_1})=v_{s_0}-v_{s_1},\;\;\forall s_0,s_1\in S.$$

We identify $\eN(x_S)$ with the subspace of $Z_S(x)\subset (T_xM)^L$ consisting of vectors $v_L=(v_\ell)_{\ell\in L}$, $v_\ell\in T_xM$ such that

$$v_\ell=0,\;\;\forall \ell\in L\setminus S,\;\; \sum_{s\in S} v_s=0. \tag{1}\label{1}$$

We have a natural $\newcommand{\bsP}{\boldsymbol{P}}$  projector

$$\bsP_S: (T_x M)^L\to Z_S(x)$$

defined as follows. For a vector $\vec{v}=(v_\ell)_{\ell\in L} \in (T_xM)^L$, we denote by $\newcommand{\bb}{\boldsymbol{b}}$ $\bb_S(\vec{v})\in T_xM$  the barycenter of its  $S$-component

$$\bb_S(\vec{v}):=\frac{1}{|S|}\sum_{s\in S} v_s\in T_x M,$$

and we set

$$\bsP_S(\vec{v}) =(\bar{v}_s)_{s\in S},\;\;\bar{v}_s:=v_s-\bb_S(\vec{v}),\;\;\forall s\in S,\;\;v_\ell=0.$$

Note that  for $u_S\in (T_xM)^S$ we have  $u_S\sim \bsP_S u_S$.

We denote by $\Bl(S,M)$ the radial blowup of $M^S$ along $\Delta_S$.  This is  manifold with boundary whose interior  is naturally identified with $M_*^S:=M^S\setminus \Delta_S$. $\newcommand{\bsS}{\boldsymbol{S}}$  Its  boundary is $\bsS(\eN)$, the "unit" sphere  bundle bundle of $\eN$.    Equivalently  we identify the fiber of $\bsS(\eN)$ at $x_s$ with the quotient

$$\bsS(\eN(x^S))=\bigl(\; Z_S(x)\setminus 0\; \bigr)/\propto,$$

where

$$u_S\propto v_S \Llra \exists c>0:\;\; v_S=c u_S.$$

Following Fulton and MacPherson we will refer to the elements  in $Z_S(x)$ as $S$-screens at $x$.   Up to  a a positive rescaling, an $S$-screen  at $x^S\in \Delta_S$ is a collection of points  $u_S\in (T_xM)^S\setminus 0^S$  with barycenter at the origin.  If we fix a metric $g$ on $M$, then the fiber $\bsS\bigl(\;\eN(x_s)\;\bigr)$ that can be identified with the collection $v_L\in (T_xM)^L$ satisfying (\ref{1}) and

$$\max_{s\in S}|v_s|=1. \tag{3}\label{3}$$

There is a natural   smooth surjection (blow-down map)

$$\beta_S:\Bl(S,M)\to M^S$$

whose restriction to the interior of $\Bl(S,M)$ $\DeclareMathOperator{\int}{\boldsymbol{int}}$ induces a diffeomorphism to $M^S_*$    We denote by $\beta_S^{-1}$ the inverse

$$\beta_S^{-1}: M_*^S\to \int \Bl(S,M).$$

For $x_S\in M_*^S$ we set $\newcommand{\ve}{{\varepsilon}}$

$$\hat{x}_S:=\beta_S^{-1}(x_S).$$

If

$$[0,\ve)\ni t\mapsto x_S(t) M^S,\;\; x_S(0)=x_0^S\in\Delta_S$$

is a real analytic   path such that $x_S(t)\in M_*^S$ for $t>0$, then the limit $\lim_{t\searrow 0}\hat{S}(t)$ can be described as follows.
• Fix  local (real analytic) coordinate near $x_0$ so that the points $x_{s}(t)$ can be  identified with points in a neighborhood of $0\in\bR^m$.
• For $t>0$ denote by $\bb(t)$ the barycenter of the collection $(x_s(t))_{s\in S}\subset\bR^m$, $$\bb(t)=\frac{1}{|S|}\sum_{s\in S} x_s(t).$$
• For $t>0$  and $s\in S$  define  $\bar{x}_s(t) =x_{s}(t)-\bb(t)$,  $m(t)=\max_s|\bar{x}_s(t)|$.
Then $\lim_{t\searrow 0}\hat{S}(t)$ can be identified with the vector

$$\lim_{t\searrow 0}\frac{1}{m(t)} \bigl(\;\bar{x}_s(t)_{s_\in S}\;\bigr)\in \eN(x_0^S).$$

We have a natural map $\newcommand{\eX}{\mathscr{X}}$

$$\gamma: M(L)\to \eX(M,L):=M^L\times\prod_{|S|\geq 2} \Bl(S,M),\;\; M(L)\ni x_L\mapsto \gamma(x_L):=\Bigl(\; x_L;\;\;(\hat{x}_S)_{|S|\geq 2}\;\Bigr)\in \eX(M,L).$$

The Fulton-MacPherson compactification of $M(V)$ is the closure of $\gamma\bigl(\;M(L)\;\bigr)$ in $\eX(M,L)$.

We want to give a more  explicit description of this  closure.  Observe first that $\eX(M,L)$ and thus any point in the closure of $\gamma(M[L])$ can be approached from within $\gamma(M[L])$ along a real analytic path.  Suppose  that $(0,\ve)\ni t \mapsto x_L(t)$  is a real analytic path such that $\gamma\bigl(\; x_L(t)\;\bigr)$ approaches a  point $\gamma^0\in \eX(M,L)$. The limit point is a collection $( x^*_L, (y(S))_{|S|\geq 2})\in \eX(M,L)$.

To the point $x^*_L\in M^L$ we associate an equivalence relation on $L$

$$\ell_0\sim_0 \ell_1 \Llra x^*_{\ell_0}=x^*_{\ell_1}.$$

Denote by $\newcommand{\eC}{\mathscr{C}}$ $\eC_0\subset 2^S$ the collection of equivalence classes  of $\sim_0$ of cardinality $\geq 2$.

The  subsets $S$ of $L$ of cardinality $\geq 2$  are of two types.

1.  The set $S$ is not contained in  any of the equivalence classes in $\eC_0$, i.e., $\exists$s_0,s_1\in S$such that$x^*_{s_0}\neq x^*_{s_1}$. We will refer to such subsets as separating subsets.Then$y_S=\hat{x}_S^*$. 2. The subset$S$is contained in an equivalence class$C\in \eC_0$. In other words there exists$x^*(C)\in M$such that$x^*_{s}=x^*(C)\in M$,$\forall s \in C$. We will refer to such a subset as non-separating. Then$y(S)$is an$S$-screen at$x^*(C)$,$y(S)=\bigl(\;y(S)_s\;\bigr)_{s\in S}$. Fix an equivalence class$C\in\eC_0$. Here is how one computes$y_S$for$S$non-separating,$S\subset C$. The point$x^*(C)^S\in\Delta^S$is approached along the real analytic path $$(0,\ve)\ni t\mapsto x_S(t)\in M(S).$$ Choose real analytic local coordinates at$x^*(C)$so a neighborhood of this point in$M$is identified with a neighborhood of$0$in$\bR^m$. We have Taylor expansions $$x_s(t) = v_s(1) t+v_2(2)t^2+\cdots ,\;\; s\in S.$$ For$k\geq 1$and$s\in S$we denote by$[x_s(t)]_k$the$k$-th jet of$x_s(t)$at$0$$$[x_s(t)]_k:=\sum_{j=1}^k v_s(j) t^j.$$ For each$k\geq 1$we have an equivalence relation$\sim_k$on$S$given by $$s \sim_k s'\Llra [x_s(t)]_k=[x_{s'}(t)]_k.$$. We denote by$\sim_0$the trivial equivalence relation on$C$with a single equivalence class$C$. Let$k=k_C(S)$be the smallest$k$such that$\sim_k$is a nontrivial equivalence relation on$S$. The integer$k_C(S)$is called the separation order of$S$. Then the$S$-screen$y(S)$is described as the projection $$y(S)\propto \bsP_S v_S(k),\;\; v_S(k)=\bigl(\; v_s(k)\;\bigr)_{s\in S}.$$ Remark 1. Suppose$S\subset S'\subset C\in \eC_0$and$|S|\geq 2$. Then$k_C(S) \geq k_C(S')$. Moreover $$k_C(S)=k_C(S') \Llra \bsP_Sy_{S'} \neq 0 \Llra y(S)\propto \bsP_S y(S').$$ The condition$\bsP_S y(S)\neq 0$signifies that there exist$s_0,s_1\in S'$such that $$y(S')_{s_0}\neq y(S')_{s_1}.$$ Note that if$S_0,S_1\subset C$,$|S_0|,|S_1|\geq 2$then $$k_C(S_0\cup S_1)\leq \min\bigl\lbrace\; k_C(S_0),k_C(S_1)\;\bigr\rbrace.$$ Recall that we have a sequence of equivalence relations$\sim_k$on$C\in \eC_0$. They are finer and finer$\sim_k\prec \sim_{k+1}$, i.e. $$s\sim_{k+1}s'\Rightarrow s\sim_k s'.$$ Observe that $$S\subset C,\;\;|S|\geq 2,\;\; k_C(S)>k \Llra \mbox{S is contained in an equivalence class of \sim_k.} \tag{4}\label{4}$$ Equivalently $$S\subset C,\;\;|S|\geq 2,\;\; k_C(S)\leq k \Llra \mbox{exist distinct equivalence classes S_0,S_1 of \sim_k such that}\;\; S\cap S_0, S\cap S_1\neq \emptyset \tag{4'}\label{4'}$$ Let$N_C$denote the smallest$N$such that all the equivalence classes of$\sim_N$consists of single points., i.e., $$s \sim_N s'\Llra s=s'.$$ Consider$\newcommand{\eS}{\mathscr{S}}$the collection$\eS_C$of all the equivalence classes of cardinality$\geq 2$of the relations$\sim_k$,$k\geq 0$on$C$. This is a nested family of subsets of$C$i.e., if$S_0, S_1\in\eS_C$, then $$S_0\cap S_1 \neq \emptyset \Llra S_0\subset S_1 \;\;\mbox{or}\;\;S_1\subset S_1.$$ Moreover$C\in \eS_C$. Observe that if$S_0,S_1\in \eS_C$and$S_0\subsetneq S_1$, then$k_C(S_0)> k_C(S_1)$. Using Remark 1 we deduce $$S_0,S_1\in \eS_C,\;\;S_0\subsetneq S_1 \Rightarrow \bsP_{S_0}y(S_1)=0. \tag{5}\label{5}$$ Suppose now that$S\subset C$and$|S|\geq 2$. We set $$\hat{S}=\bigcap_{S\subset S' \in\eS_C} S'.$$ In other words,$\hat{S}$is the smallest subset in$\eS_C$containing$S$. Lemma 2. We have$k_C(S)= k_C(\hat{S})$. Proof. Observe first we have$k_C(S)\leq k_C(\hat{S})$. Set$k_0 :=k_C(S)$. If$k_C(\hat{S})> k_0$, then (\ref{4}) implies$\hat{S}$is contained in an equivalence class of$\sim_{k_0}$. On the other hand$k_C(S)=k_0$(\ref{4'}) implies$S_0$intersects nontrivially two equivalence classes of$\sim_{k_0}$. This contradicts the condition$S\subset \hat{S}$. qed Using Remark 1 we deduce $$S\subset C,\;\;|S|\geq 2 \Rightarrow y(S)\propto \bsP_S y(\hat{S}). \tag{6}\label{6}$$ The conditions (\ref{5}), (\ref{6}) describe some compatibility conditions satisfied by the screens$y(S)$,$S\subset L$non-separating. We can now form the family of subsets of$L$$$\eS=\bigcup_{C\in\eC_0} \eS_C.$$ This also a nested family of subsets of cardinality$\geq 2$. A subset$S\subset L$of cardinality$\geq  2$is called$\eS$-separating if it is not contained in any of the sets of$\eS$. Otherwise it is called nonseparating. For any separating set$S$we denote by$\hat{S}$the smallest subset of$\eS$containg$S$. The limit point $$c:= \Bigl(\;x^*(L), \bigl(\;y(S)\;\bigr)_{S\subset L,\;|S|\geq 2}\;\Bigr)\in\eX(M,L)$$ satifies the following conditions. $$y(S)\in \beta^{-1}_S\bigl(\;M^S_*\;\bigr),\;\; \mbox{if S is separating}. \tag{C_1} \label{C1}$$ $$y(S) \;\;\mbox{is an S-screen if S is non-separating}. \tag{C_2} \label{C2}$$ $$S_0,S_1\in \eS,\;\;S_0\subset S_1\Rightarrow \bsP_{S_0}y(S_1)=0. \tag{C_3}\label{C3}$$ $$S\;\;\mbox{nonseparating} \Rightarrow y(S)\propto \bsP_S y(\hat{S}). \tag{C_4}\label{C4}$$ Comments. (a) Let us recall that (\ref{C3}) signifies that the components$y(S_1)_ss\in S_0$are identical. (b) Let me say a few words about the interpretation of the nested family$\eS$. A set$S$corresponds to a collection of distinct points in$(x_s)_{s\in S}$in$M$that is clustering ner a point$x^*$. A subset$S'$corresponds to a subcollection of the above collection that is clustering at a faster rate. Running the above arguments in revers one can show that a collection $$\Bigl(\;x^*(L), \bigl(\;y(S)\;\bigr)_{S\subset L,\;|S|\geq 2}\;\Bigr)\in\eX(M,L)$$ belongs to the closure of$\gamma\bigl(\;M(L)\;\bigr)$in$\eX(M, L)$if and only if there exists a nested collection$\eS$of subsets of$L$of cardinality$\geq 2$such that satisfying the compatibility conditions (\ref{C1}-\ref{C4}) are satisfied. The set$\eS$is called the type of the limit point. For a nested family$\eS$of subsets of$L$of cardinality$\geq 2$we denote Define$M^(\eS)$the collection of points of type$\eS$. The stratum$M(\eS)$has codimension$|\eS|$. This can be seen after a tedious computation that takes into account a (\ref{C1}-\ref{C4}) . To explain introduce a notation. Given$S, S'\in \eS$we say that$S$precedes$S'$and we write this$S\lessdot S'$, if$S$is maximal amomgst the subsets of$\eS$contained but not equal to$S'$. Denote by$\eS_{\max}$the collection of maximal sets in$\eS$. (The collection$\eS_{\max}$coincides with the collection$\eC_0$in the above discussion.) The, if we recall that$\dim M=m$and$|L|=n$we deduce $$\dim M(\eS)^* =m\left(\; n-\sum_{S\in\eS_{\max}}(|S|-1)\;\right) +\sum_{S\in \eS}\left[\;\;m\left(\;(|S|-1)-\sum_{S'\lessdot S}\bigl(\;|S'|-1\;\bigr)\right)-1\;\right]$$ To understand this formula let us consider a point $$c =\Bigl(\;x^*(L), \bigl(\;y(S)\;\bigr)_{S\subset L,\;|S|\geq 2}\;\Bigr)\in M(\eS)^*.$$ The coordinates of$x^*(L)$are described by$nm$parameters Each$S\in \eS_{\max}$introduces the constraints $$x^*(L)_{s_1}=x^*(L)_{s_2},\forall s_1,s_2\in S.$$ If$S=\lbrace s_1,\dotsc,s_N\rbrace$we see that the above constraints are consequences of the linearly independent ones $$x^*(L)_{s_1}-x^*(L)_{s_2}= \cdots =x^*(L)_{s_{N-1}}-x^*(L)_{s_N}=0.$$ These cut down the number of parameters required to describe$x^*(L)$by$m(N-1)=m(|S|-1)$. Thus the number of parameters need to describe$x^*(L)$is$m\left(\; n-\sum_{S\in\eS_{\max}}(|S|-1)\;\right)$From (\ref{C3}) and (\ref{C4}) we deduce that the collection $$\bigl(\;y(S)\;\bigr)_{S\subset L,\;|S|\geq 2}$$ is uniquely determined by the subcollection $$\bigl(\;y(S)\;\bigr)_{S\in\eS} .$$ The screen$y(S)$belongs to the unit sphere$\bsS(\eN(x_S))$which has dimension $$\dim M^S-\dim\delta_S-1= m(|S|-1)-1.$$ Thus we need$m(|S|-1)$parameters to describe the screen$y(S)$. However, the condition (\ref{C3}) shows that any$S'\lessdot S$induces$m(|S'|-1)$linearly independent constraints on these parameters so that$y(S)$has a total of $$m\left(\;(|S|-1)-\sum_{S'\lessdot S}\bigl(\;|S'|-1\;\bigr)\right)-1$$ degrees of freedom. We want to describe a neighborhood of$M(\eS)$in$M[L]$. We will achieve this via an explicit map $$\Psi : M(\eS)\times \bR_{\geq 0}^{\eS} \to M[L]$$ defined as follows. Denote by$\vec{t}=(t_S)_{s\in\eS}$the coordinates on$\bR^{\eS}_{\geq 0}$. For$S\in \eS$we set $$T_S=\prod_{\eS\ni S'\supseteq S} t_{S'}.$$ If $$c = (x(c), (y(S,c))_{S\in\eS})\in M(\eS),$$ then $$\Psi(c, \vec{t})= \bigl( x_\ell (c,\vec{t})\;\bigr)_{\ell \in L},$$ where $$x_\ell(c,\vec{t})= x(c)_\ell +\sum_{\ell\in S\in \eS} T_S y(S,c)_\ell.$$ In the above formula$y(S)$is assumed to be a vector of norm$1$in$Z_S(x_\ell)$. Let us convince ourselves that for fixed$c_0\in M(\eS)$there exists a small neighborhood$U$of$c_0$in$M(\eS)$and a neighborhood$V$of$0\in\bR^{\eS}_{\geq 0}$such that$\Psi$maps$U\times  V_{>0}$into$M(L)$. Here$V_{>0}-V\cap \bR^{\eS}_{>0}$. Thus we have to show that if$i,j\in L$,$i\neq j$, then for$c$close to$c_0$and$\vec{t}$close to$0$. $$x_i(c,\vec{t})\neq x_j(c,\vec{t})$$ Note that a set$S\in\eS$that contains$i$is either contained in$S_0$or contains$S_0$. A similar fact is true for$j$. Observe that if$S\supset\neq S_0$then$y(S,c)_i=y(S,c)j$. Thus $$x(c,\vec{t})_i-x(c,\vec{t})_j =\sum_{S\subsetneq S_0} T_S\bigl(\; y(S)_i-y(S)_j\;\bigr)+ T_{S_0}(y(S_0)_i-y(S_0)_j$$ $$=T_{S_0}\left(\sum_{S\subsetneq S_0} \tau _S\bigl(\; y(S)_i-y(S)_j\;\bigr)+ (y(S_0)_i-y(S_0)_j\;\right),$$ where $$\tau_S=\prod_{S\subset S'\subset\neq S_0} t_{S'}.$$ The conclusion follows by observing that$y(S)_i\neq y(S)_j$. We denote by$M[\eS]$the closure of$M(\eS)$in$M[L]\$. Observe that

$$M(\eS')\subsetneq M[\eS] \Llra \eS'\supsetneq \eS.$$