A Day of Triangulations. A conference around Manolescu's new result

A Day of Triangulations

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 $M^S$. We can identify $x^S$ with a point in $x\in M$ so

$$T_{x^S}M^S\cong (T_xM)^S.$$

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 $\eN_S(x)$ of the normal bundle  $\eN_S=T_{x^S}M^S/T_{x^S}\Delta_S$ at a point $x^S$ is the quotient of $(T_xM)^S$ modulo the equivalence relation $\newcommand{\Llra}{\Longleftrightarrow}$

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

We identify $\eN_S(x)$ 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

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

is a real analytic   path such that $x_S(t)\in M_*^S$ for $t>0$, then the limit $\lim_{t\searrow 0}\hat{x}_S(t)$ 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 $\lim_{t\searrow 0}\hat{x}_S(t)$ 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 $y(S_1)_s$ $s\in S_0$ 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

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