MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES
DEV P. SINHA
Abstract.We present new definitions for and give a comprehensive treatmen*
*t of the canonical compact-
ification of configuration spaces due to Fulton-MacPherson and Axelrod-Si*
*nger in the setting of smooth
manifolds, as well as a simplicial variant of this compactification. Our *
*constructions are elementary and
give simple global coordinates for the compactified configuration space o*
*f a general manifold embedded
in Euclidean space. We stratify the canonical compactification, identifyi*
*ng the diffeomorphism types of
the strata in terms of spaces of configurations in the tangent bundle, an*
*d give completely explicit local
coordinates around the strata as needed to define a manifold with corners*
*. We analyze the quotient map
from the canonical to the simplicial compactification, showing it is a ho*
*motopy equivalence. We define
projection maps and diagonal maps, which for the simplicial variant satis*
*fy cosimplicial identities.
Contents
1. Introduction *
* 1
1.1. The basic definitions *
* 2
1.2. Review of previous work *
* 2
1.3. A comment on notation, and a little lemma *
* 3
1.4. Acknowledgements *
* 3
2. A category of trees and related categories *
* 3
3. The stratification of the basic compactification *
* 6
3.1. Stratification of Cn[M] using coordinates in An[M] *
* 6
3.2. Statement of the main theorem *
* 6
3.3. The auxilliary construction, Cn{Rk} *
* 8
3.4. Proof of Theorem 3.8 for M = Rk *
* 10
3.5. Proof of Theorem 3.8 for general M *
* 12
4. First properties *
* 13
4.1. Characterization in An[M] and standard projections *
* 13
4.2. Manifold structure, codimensions of strata, functoriality for embedding*
*s, and equivariance 14
4.3. The closures of strata *
* 15
4.4. Configurations in the line and associahedra *
* 16
5. The simplicial compactification *
* 18
6. Diagonal and projection maps *
* 21
References *
* 24
1.Introduction
Configuration spaces are fundamental objects of study in geometry and topolog*
*y, and over the past
ten years, functorial compactifications of configuration spaces have been an im*
*portant technical tool. We
review the state of this active area after giving our definitions.
___________
1991 Mathematics Subject Classification. Primary: 55T99.
1
2 DEV P. SINHA
1.1. The basic definitions. We deal with products of spaces extensively, so we *
*first set down some
efficient notation to manage products.
Notation. If S is a finite set, XS is the product X#S where #S is theQcardinali*
*ty of S. Consistent with
this, if {Xs} is a collection of spaces indexed by S, we let (Xs)S = s2SXs. F*
*or coordinatesQin either
case we use (xs)s2S or just (xs) when S is understood. Similarly, a product of *
*maps s2Sfs may be
written(fs)s2Sor just (fs). We let n_denote the set {1, . .,.n}, our most commo*
*n indexing set.
Definition 1.1. If M is a smooth manifold and let Cn(M) be the subspace of (xi)*
* 2 Mn_such that xi6= xj
if i 6= j. Let ' denote the inclusion of Cn(M) in Mn_.
Suppose that M were equipped with a metric. The main compactification which w*
*e study, Cn[M], is
homeomorphic to the subspace of Cn(M) for which d(xi, xj) ffl for some suffic*
*iently small ffl. From this
model, however, it is not clear how Cn(M) should be a subspace of the compactif*
*ication, much less how
to establish functorality or more delicate properties we will develop.
Definition 1.2. For (i, j) 2 C2(n_), let ßij:Cn(Rk) ! Sk-1 be the map which sen*
*ds (xi) to the unit
vector in the direction of xi- xj. Let I be the closed interval from 0 to 1, an*
*d for (i, j, k) 2 C3(n_) let
sijk:Cn(Rk) ! I = [0, 1] be the map which sends (xi) to (|xi- xj|=|xi- xk|).
Our compactifications are defined as closures, for which we also set notation.
Notation. If A is a subspace of X we let clX (A), or simply cl(A) if by context*
* X is understood, denote
the closure of A in X.
From now on by a manifold M we mean a submanifold of some Rk, so that Cn(M) i*
*s a submanifold of
Cn(Rk). For M = Rk, we specify that Rk is a submanifold of itself through the i*
*dentity map.
Definition 1.3. Let An[M], the main ambient space in which we work, be the prod*
*uct Mn_x(Sk-1)C2(n_)x
IC3(n_), and similarly let An<[M]> = Mn_x (Sk-1)C2(n_). Let
ffn = ' x ßij|Cn(M)x (sijk)|Cn(M): Cn(M) ! An[M]
and define Cn[M] to be clAn[M](im(ffn)). Similarly, let fin = ' x ßij|Cn(M): Cn*
*(M) ! An<[M]> and
define Cn<[M]> to be clAn<[M]>(im(fin)).
We will show that Cn[M] is a manifold with corners whose diffeomorphism type *
*depends only on that
of M below. Because An[M] is compact when M is, and Cn[M] is closed in An[M], w*
*e immediately have
the following.
Proposition 1.4. If M is compact, Cn[M] is compact.
We call Cn[M] the canonical compactification of Cn(M) and Cn<[M]> the simplic*
*ial variant. When M
is not compact but is equipped with a complete metric, it is natural to call Cn*
*[M] the canonical completion
of Cn(M).
1.2. Review of previous work. The compactification Cn[M] first appeared in work*
* of Axelrod and
Singer [1], who translated the definition of Fulton and MacPherson in [10] as a*
* closure in a product of
blow-ups from algebraic geometry to the setting of manifolds using spherical bl*
*ow-ups. Kontsevich made
similar constructions at about the same time as Fulton and MacPherson, and his *
*later definition in [15]
coincides with our eCn<[Rk]>, though it seems that he was trying to define eCn[*
*Rk]. Kontsevich's oversight
was corrected in [11], in which Gaiffi gives a definition of Cn[Rk] similar to *
*ours, generalizes the construction
for arbitrary hyperplane arrangments over the real numbers, gives a pleasant de*
*scription of the category
of strata using the language of blow-ups of posets from [9], and also treats bl*
*ow-ups for stratified spaces
locally and so gives rise to a new definition of Cn[M].
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 3
Axelrod and Singer used these compactifications to define invariants of three*
*-manifolds coming from
Chern-Simons theory, and these constructions have generally been vital in quant*
*um topology [3, 17, 2, 19].
Extensive use of similar constructions has been made in the setting of hyperpla*
*ne arrangments [6, 25]
over the complex numbers. These compactifications have also inspired new comput*
*ational results [16, 24],
and they canonically realize the homology of Cn(Rk) [21]. We came to the presen*
*t definitions of these
compactifications so we could define maps and boundary conditions needed for ap*
*plications to knot theory
[4, 22].
New results include full proofs of many folk theorems, and the following:
oA construction for general manifolds which bypasses the need for blow-ups*
*, uses simple global
coordinates, and through which functorality is immediate.
oExplicit description of the strata in terms of spaces of configurations i*
*n the tangent bundle.
oFull treatment of the simplicial variant, including a proof that the proj*
*ection from the canonical
compactification to the simplicial one is a homotopy equivalence.
oA clarification of the central role which Stasheff's associahedron plays *
*in this setting.
oConstructions of diagonal maps, projections, and substitution maps as nee*
*ded for applications.
The constructions of these maps are signifcantly aided by having simple g*
*lobal coordinates.
In future work [18], we will use these constructions to define an operad stru*
*cture on these compactifica-
tions of configurations in Euclidean space, which has consequences in knot theo*
*ry. This operad structure
was first applied in [12].
We also hope that a unified and explicit exposition of these compactification*
*s using our simplified
definition could be of help, especially to those who are new to the subject.
1.3. A comment on notation, and a little lemma. There are two lines of notation*
* for configuration
spaces of manifolds in the literature, namely Cn(M) and F(M, n). Persuaded by B*
*ott, we choose to use
the Cn(M) notation. Note, however, that Cn(M) in this paper is C0n(M) in [3] an*
*d that Cn[M] in this
paper is Cn(M) in [3]. Indeed, we warn the reader to pay close attention to the*
* parentheses in our notation:
Cn(M) is the open configuration space; Cn[M] is the Fulton-MacPherson/Axelrod-S*
*inger compactification,
its canonical completion; Cn<[M]>, the simplicial variant, is a quotient of Cn[*
*M]; Cn{M}, an auxilliary
construction, is a subspace of Cn[M] containing only one additional stratum. We*
* suggest that those who
choose to use F(M, n) for the open configuration space use F[M, n] for the comp*
*actification.
As closures are a central part of our definitions, we need a lemma from point*
*-set topology that open
maps commute with taking closures.
Lemma 1.5. Let A be a subspace of X, and let ß : X ! Y be an open map. Then ß(c*
*lX (A)) clY(ß(A)).
If clX (A) is compact (for example, when X is) then this inclusion is an equali*
*ty.
Proof.First, ß-1(clY(ß(A))) is closed in X and contains A, so it contains clX (*
*A) as well. Applying ß to
this containment we see that ß(clX (A)) clY(ß(A)).
If clX (A) is compact, so is ß(clX (A)), which is thus closed in Y . It conta*
*ins ß(A), therefore clY(ß(A))
ß(clX (A)).
1.4. Acknowledgements. The author would like to thank Dan Dugger for providing *
*a proof and refer-
ences for Lemma 5.6, Ismar Volic for working with the author on an early draft *
*of this paper, Matt Miller
for a careful reading, and Giovanni Gaiffi and Eva-Maria Feitchner for sharing *
*preprints of their work.
2.A category of trees and related categories
In order to understand the compactifications Cn[M] we have to understand thei*
*r strata, which are
naturally labelled by a poset (or category) of trees.
4 DEV P. SINHA
Definition 2.1. Define an f-tree to be a rooted, connected tree, with labelled *
*leaves, and with no bivalent
internal vertices. Thus, an f-tree T is a connected acyclic graph with a specif*
*ied vertex v0 called the root.
The root may have any valence, but other vertices may not be bivalent. The univ*
*alent vertices other than
perhaps the root are called leaves, and each leaf is labelled uniquely with an *
*element of #l(T)_, where l(T)
is the set of leavesPofSTfandr#l(T)aisgitsrcardinality.eplacements_
Figure12.2.A tree T.
2
3
4
5
6
7
v1
v0
v2
v3
In an f-tree there is a unique path from any vertex or edge to the root verte*
*x, which we call its root
path. We say that one vertex or edge lies over another if the latter is in the *
*root path of the former. For
any edge, its boundary vertex closer to the root is called its initial vertex, *
*and its other vertex is called its
terminal vertex. If two edges share the same initial vertex we call them coinci*
*dent. For a vertex v there is
a canonical ordering of edges for which v is initial, the collection of which w*
*e call E(v), the group of edges
coincident at v. Namely, set e < f if the smallest label for a leaf over e is s*
*maller than that over f. We
may use this ordering to name these edges e1(v), . .,.e#v(v), where #v is the n*
*umber of edges f in E(v).
We will be interested in the set of f-trees as a set of objects in a category*
* in which morphisms are
defined by contracting edges.
Definition 2.3. Given an f-tree T and a set of non-root, non-leaf edges E the c*
*ontraction of T by E
is the tree T0 obtained by, for each edge e 2 E, identifying its initial vertex*
* with its terminal vertex and
removing e from the set of edges.
Definition 2.4. Define n_to be the category whose objects are f-trees with n l*
*eaves. There is a (unique)
morphism in n_from T to T0 if T0 is isomorphic to a contraction of T along som*
*e set of edges.
Figure 2.5.The category 3.
Finally, let V (T) denote the set of non-leaf vertices of T. Let V i(T) denot*
*e its subset of internal vertices
(thus only excluding the root). Note that a morphism in n_decreases the number*
* of internal vertices,
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 5
which is zero for the terminal object in n_. Let f nbe the full subcategory o*
*f f-trees whose root is
univalent (informally, trees with a trunk). Note that f nhas an operad structur*
*e, as defined in [12].
It is useful to have facility with categories which are essentially equivalen*
*t to n_. We will define these
categories through the notions of parenthesization and exclusion relation. Furt*
*her equivalent constructions
include the collections of screens of Fulton and MacPherson [10]. The best pers*
*pective on these categories
is given by the combinatorial blow-up of Feitchner and Kozlov [9]. Indeed, Gaif*
*fi shows in [11] that the
poset of strata of a blow-up of an arrangment is the combinatorial blow-up of t*
*he orignial poset associated
to the arrangment. Since we focus not on general blow-ups but on compactified c*
*onfiguration spaces in
particular, we choose more concrete manifestations of this category.
Definition 2.6. A (partial) parenthesization P of a set S is a collection {Aff}*
* of nested subsets of S, each
of cardinality greater than one. By nested we mean that for any ff, fi the inte*
*rsection Aff\Afiis either Aff,
Afior empty. The parenthesizations of S form a poset, which we call Pa(S), in w*
*hich P P0if P P0.
Parenthesizations are related to trees in that they may keep track of sets of*
* leaves which lie over the
vertices of a tree.
Definition 2.7. Define f1: n_! Pa(n_) by sending a tree T to the collection of *
*sets {Av}, where
v 2 V i(T) and Av is the set of indices of leaves which lie over v. Define g1:P*
*a(n_) ! n_by sending a
parenthesization to a tree with the following data
oOne internal vertex vfffor each Aff.
oAn edge between vffand vfiif Aff Afibut there is no proper Aff Afl Afi.
oA root vertex with an edges connecting it to each internal vertex corresp*
*onding to a maximal Aff.
oLeaves with labels in n_with an edge connecting the ith leaf to either th*
*e vertex vffwhere Affis is
the minimal set containing i, or the root vertex of there is no such Aff.
We leave to the reader the straightforward verification that f1 and g1 are we*
*ll-defined and that the
following proposition holds.
Proposition 2.8. The functors f1 and g1 are isomorphisms between the categories*
* n_and Pa(n_).
Another way in which to account for the data of which leaves lie above common*
* vertices in a tree is
through the notion of an exclusion relation.
Definition 2.9. Define an exclusion relation R on a set S to be a subset of C3(*
*S) such that the following
properties hold
(1)If (x, y), z 2 R then (y, x), z 2 R and (x, z), y =2R.
(2)If (x, y), z 2 R and (w, x), y 2 R then (w, x), z 2 R.
Let Ex(S) denote the poset of exclusion relations on S, where the ordering is d*
*efined by inclusion as
subsets of C3(S).
We now construct exclusion relations from parenthesizations, and vice versa.
Definition 2.10. Let f2:Pa(n_) ! Ex(n_) be defined by setting (i, j), k 2 R if *
*i, j 2 Affbut k =2Aff
for some Affin the given parenthesization. Define g2:Ex(n_) ! Pa(n_) by, given *
*an exclusion relation R,
taking the collection of sets A~i,:kwhere A~i,:kis the set of all j such that (*
*i, j), k 2 R, along with i
when there is such a j. Let Tr = g1O g2 : Ex(n_) ! n_and let Ex= f2O f1.
As above, we leave the proof of the following elementary proposition to the r*
*eader.
Proposition 2.11. The composite f2O g2 is the identity functor. If f2(P) = f2(P*
*0) then P and P0may
only differ by whether or not they contain the set n_itself.
6 DEV P. SINHA
3. The stratification of the basic compactification
This section is the keystone of the paper. We first define a stratification o*
*f Cn[M] through coordinates
as a subspace of An[M]. For our purposes, a stratification is any expression of*
* a space as a finite disjoint
union of locally closed subspaces called strata, which are usually manifolds, s*
*uch that the closure of each
stratum is its union with other strata. We will show that when M has no boundar*
*y, the stratification
we define through coordinates coincides with the stratification of Cn[M] as a m*
*anifold with corners. The
strata of Cn[M] are individually simple to describe, so constructions and maps *
*on Cn[M] are often best
understood in terms of these strata.
Before treating Cn[M] in general, we would like to be completely explicit abo*
*ut the simplest possible
case, essentially C2[Rk].
Example. Let C*2(Rk) ~=Rk - 0 be the subspace of points (0, x 6= 0) 2 C2(Rk) an*
*d consider its closure
as the subspace of Rkx Sk-1 of points (x 6= 0, _x_||x||). The projection of thi*
*s subspace onto Sk-1 coincides
with the tautological positive ray bundle over Sk-1, which is a trivial bundle.*
* The closure C*2[Rk] is the
non-negative ray bundle, which is diffeomorphic to Sk-1x [0, 1). Projecting thi*
*s closure onto Rk is a
homeomorphism when restricted to Rk- 0, and the preimage of 0 is a copy of Sk-1*
*, the stratum of added
points. Thus, C*2[Rk] is diffeomorphic to the blow-up of Rk at 0, in which one *
*replaces 0 by the sphere of
directions from which it can be approached. Unlike the locally-defined blow-up,*
* C*2[Rk] has simple global
coordinates inherited from Rkx Sk-1.
3.1. Stratification of Cn[M] using coordinates in An[M]. We proceed to define a*
* stratification for
Cn[M] by associating an f-tree to each point in Cn[M].
Definition 3.1. Let x = ((xi), (uij), (dijk))2 Cn[M]. Let R(x) be the exclusio*
*n relation defined by
(i, j), k 2 R(x) if dijk= 0. Let T(x) be equal to either Tr(R(x)) or, if all of*
* the xiare equal, the f-tree
obtained by adding a new root to Tr(R(x)).
Note that because dijkdi`j= di`kfor points in the image of Cn(M), by continui*
*ty this is true for all
of Cn[M]. So if dijk= 0 = di`jthen di`k= 0. Therefore, R(x) satisfies the last *
*axiom for an exclusion
relation. The other axiom is similarly straightforward to check to see that R(x*
*) is well-defined.
Definition 3.2. Let CT(M) denote the subspace of all x 2 Cn[M] such that T(x) =*
* T, and let CT[M] be
its closure in Cn[M].
The following proposition, which gives a first indication of how the CT(M) fi*
*t together, is an immediate
consequence of the definitions above.
Proposition 3.3. Let s = {(xi)j}1j=1be a sequence of of points in Cn(M) which c*
*onverges to a point in
Cn[M] An[M]. The limit of s is in CT[M] if and only if the limit of d(xi, xj)*
*=d(xi, xk) approaches zero
for every (i, j), k 2 Ex(T) and, in the case where the root valence of T is one*
*, we also have that all of the
xiapproach the same point in M.
To a stratification of a space, one may associate a poset in which stratum ff*
* is less than stratum fi if ff
is contained in the closure of fi.
Theorem 3.4. The poset associated to the stratification of Cn[M] by the CT(M) i*
*s isomorphic to n_.
Proof.This theorem follows from the preceding proposition and the fact that if *
*T ! T0 is a morphism in
n_then R(T0) is contained in R(T).
3.2. Statement of the main theorem. Having established an intrinsic definition *
*for the CT(M) and
a combinatorial description of how they fit together, we now set ourselves to t*
*he more difficult task of
identifying these spaces explicitly. We describe the spaces CT(M) in terms of "*
*infinitesimal configurations".
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 7
Definition 3.5. oLet Simk be the subgroup of the group of affine transforma*
*tions in Rk generated
by translation and scaling.
oDefine ICi(M) to be the space of i distinct points in TM all lying in one*
* fiber, modulo the action
of Simk in that fiber. Let p be the projection of ICi(M) onto M.
For example IC2(M) is diffeomorphic to STM, the unit tangent bundle of M.
Let e 2 E0 = E(v0) be a root edge of an f-tree T, and let V (e) V i(T) be t*
*he set of internal vertices
which lie over e.
Definition 3.6. (1)Define ICe(M) to be subspace of the product (IC#v(M))V (e*
*)of tuples of infini-
tesimal configurations all sitting over the same point in M.
(2)Let pe be the map from ICe(M) onto M defined projecting onto that point.
(3)Let DT(M) be the subspace of (ICe(M))E0 of points whose image under (pe) *
*in (M)E0 sits in
C#v0(M).
In other words, a point in DT(M) is a collection of #v0 distinct points (xe)e*
*2E0in M with a collection
of #v(e) infinitesimal configurations at each xe.
Figure 3.7.A point in DT(M) with T from Figure 2.2.
The following theorem is the main theorem of this section.
Theorem 3.8. CT(M) is diffeomorphic to DT(M).
Remark. To intuitively understand CT(M) as part of the boundary of Cn[M] one vi*
*ews an element of
ICi(M) as a limit of a sequence in Ci(M) which approaches a point (x, x, . .x.)*
* in the (thin) diagonal of
Mi. Eventually, in such a sequence all the points in a configuration would lie *
*in a coordinate neighborhood
of x, which can through the exponential map can be identified with TxM, and the*
* limit is taken in that
tangent space up to rescaling. If i > 2, ICi(M) is itself not complete, so one *
*allows these infinitesimal
configurations to degenerate as well, and this is how the situation is pictured*
* in Figure 3.7. Because
T(TM) ~= 3TM, the recursive nature of having sub-configurations degenerate is n*
*ot reflected in the
topology of DT(M).
To establish this theorem we focus on the case in which M is Euclidean space *
*Rk, as DT(Rk) admits a
simple description.
Definition 3.9. Let eCn(Rk) be the quotient of Cn(Rk) by Simk acting diagonally*
*, and let q denote the
quotient map. Choose coset representatives to identify eCn(Rk) with a subspace *
*of Cn(Rk), namely the
subspace of points (xi) whose center of mass is ~0and such that the maximum of *
*the d(xi,~0) is one.
Because the tangent bundle of Rk is trivial, ICi(Rk) ~=Rkx eCi(Rk), and we ha*
*ve the following.
8 DEV P. SINHA
i Vji(T)
Proposition 3.10. DT(Rk) = C#v0(Rk) x eC#v(Rk) .
Alternately, DT(Rk) is the space in which each edge in T is assigned a point *
*in Rk, with coincident
edges assigned distinct points, modulo translation and scaling of coincident gr*
*oups of edges.
Roughly speaking, the proof of Theorem 3.8 when M = Rk respects the product d*
*ecomposition of
Proposition 3.10. We start by addressing the stratum associated to the tree *
* with a single internal
vertex connected to a univalent root.
3.3. The auxilliary construction, Cn{Rk}.
Definition 3.11. Let An{M} = (M)n_x (Sk-1)C2(n_)x (0, 1)C3(n_), a subspace of A*
*n[M]. Note that the
image of ffn:Cn(M) ! An[M] lies in An{M}. Let Cn{M} be clAn{M}(im(ffn)).
For our purposes, Cn{M} will be useful as a subspace of Cn[M] to first unders*
*tand, which we do for
M = Rk.
Theorem 3.12. Cn{Rk} is diffeomorphic to Dn{Rk} = Rkx eCn(Rk) x [0, 1).
As a manifold with boundar Cn{Rk} has two strata, namely RkxCen(Rk)x(0, 1), w*
*hich we will identify
with Cn(Rk), and Rkx eCn(Rk) x 0, the points added in this closure. We will see*
* that these correspond to
C (Rk) and C (Rk) respectively.
To prove Theorem 3.12 we define a map :Dn{Rk} ! An{Rk} and show that it is *
*a homeomorphism
onto Cn{Rk}. The map will essentially be an expansion from the point in Rk o*
*f the infinitesimal
configuration given by the point in eCn(Rk) (Rk)n_.
Definition 3.13. (1)Define j :Dn{Rk} ! (Rk)n by sending x x (yi) x t to (x +*
* tyi).
(2)Let p denote the projection from Dn{Rk} onto eCn(Rk).
(3)Let eßijand esijkdenote the maps on eCn(Rk) which when composed with q gi*
*ve the original ßij
and sijk.
(4)Finally, define :Dn{Rk} ! An{Rk} by j x (eßijO p) x (esijkO p).
When t > 0, the image of j is in Cn(Rk), and moreover we have the following.
Proposition 3.14. The map |t>0coincides with ffn O j, a diffeomorphism from Rk*
* x eCn(Rk) x (0, 1)
onto the image of ffn.
Proof.For t > 0, the map j satisfies eßijO p = ßijO j, and similarly esijkO p =*
* sijkO j, showing that |t>0
coincides with ffn O j.
The inverse to |t>0is the product of: the map which sends (xi) to its the ce*
*nter of mass, the quotient
map q to eCn(Rk), and the map whose value is the greatest distance from one of *
*the xito the center of
mass. Both |t>0and its inverse are clearly smooth.
Corollary 3.15. |t=0has image in Cn{Rk}.
We come to the heart of the matter. Because Cn{Rk} is defined as a closure, t*
*o identify it more explicitly
we must identify a closed subset of An{Rk}, which we do presently. We will appl*
*y this case repeatedly in
analysis of Cn[Rk].
Definition 3.16. Let eAn[Rk] = (Sk-1)C2(n_)x IC3(n_), and let eAn{Rk} = (Sk-1)C*
*2(n_)x (0, 1)C3(n_).
Convention. We extend multiplication on (0, 1) to its closure by setting a.1 = *
*1 if a 6= 0 and 0.1 = 1.
Lemma 3.17. The map 'n = (eßij) x (esijk) : eCn(Rk) ! eAn[Rk] is a diffeomorphi*
*sm onto its image, which
is closed as a subspace of eAn{Rk}.
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 9
Proof.Collinear configurations up to translation and scaling are cleary determi*
*ned by their image under
one eßijand the esijk. For non-collinear configurations, we may reconstruct x= *
*(xi) from the uij= eßij(x)
and dijk= esijk(x) up to translation and scaling by for example setting x1 = ~0*
*, x2 = u12 and then
xi= d1i2u1ifor any i. These assigments of xiare smooth functions, so in fact 'n*
* is a diffeomorphism onto
its image.
For the sake of showing that the image of 'n is closed, as well as use in sec*
*tion 5, we note that d1i2can
be determined from the uijby the law of sines. If uij, ujkand uikare distinc*
*t then,
___|xi-_xj|__p_= ___|xj-_xk|__p_= ___|xi-_xk|__p_.
1 - (uki. ukj)2 1 - (uij. uik)2 1 - (uji. ujk)2
q _________ 2
Thus, in most cases d1i2= 1-(u2i.u21)_1-(ui1.ui2)2. In general, as long as no*
*t all points are collinear, the law of
sines above can be used repeatedly to determine all dijkfrom the uij, which sho*
*ws that when restricted
to non-collinear configurations, (eßij) itself is injective.
We identify the image of 'n as the set of all points (uij) x (dijk) which sat*
*isfy the following conditions
needed to consistently define an inverse to 'n:
(1)uij= -uji.
(2)uij, ujkand uikall lie in the same great circleqon_Sk-1,_with uikstrictly*
* between uijand ujk.
2
(3)If uij, ujkand uikare distinct then dijk= 1-(uik.ujk)_1-(uij.ujk)2
(4)dijkare non-zero and finite and
dijkdikj= 1 = dijkdjkidkij= dijkdi`jdik`.
We say a condition is closed if the subspace of points which satisfy it is cl*
*osed. Note that condition 4
follows from condition 3 when the latter applies.
Condition 1 is clearly closed, and condition 4 is a closed condition in eAn{R*
*k}, since we are already
assuming that dijk2 (0, 1). Condition 3 says that on an open subspace of this i*
*mage, the dijkare a
function of the uijand gives no restrictions away from this subspace, and so is*
* also a closed condition.
Considering condition 2, it is a closed condition for uij, ujkand uikto all lie*
* on a great circle. It is not
usually a closed condition for uikto be strictly between uijand ujk. But by con*
*dition 3, if uij6= ujk
but uik= ujk then dijk= 0, so in fact condition 2 is closed within the points i*
*n eAn{Rk} satisfying
condition 3.
Because |t=0is the product of the diagonal map Rk ! (Rk)n_, which is a diffe*
*omorphism onto its image,
with 'n we may deduce the following.
Corollary 3.18. |t=0is a diffeomorphism onto its image.
We may now finish analysis of Cn{Rk}.
Proof of Theorem 3.12.Proposition 3.14 and Corollary 3.18 combine to give that *
* T :DT{Rk} ! CT{Rk}
is injective. We thus want to show that it is surjective and has a continuous i*
*nverse.
Consider the projection p from Cn{Rk} An{Rk} to (xi) 2 (Rk)n_. Over Cn(Rk)*
* the image of ffn
is its graph, which is locally closed, so p-1(Cn(Rk)) ~=Cn(Rk). If xi= xj but x*
*i6= xk for some i, j, k,
continuity of sijkwould force dijk= 0, which is not possible in An{Rk}. Thus no*
* points in Cn{Rk} lie
over such (xi). Over the diagonal of (Rk)n_we know that Cn{Rk} contains at leas*
*t the image of |t=0. But
by Lemma 3.17, we may deduce that this image is closed in An{Rk} and thus accou*
*nts for all of Cn{Rk}
over the diagonal.
We define an inverse to T according to this decomposition over (Rk)n_. For a*
* point in Cn(Rk), the
inverse was given in Proposition 3.14. For points over the diagonal (xi = x) in*
* (Rk)n_, the inverse is a
10 DEV P. SINHA
product of: the map which sends such a point to x 2 Rk, '-1n, and the constant *
*map whose image is
0 2 [0, 1). Smoothness of this inverse is straightforward and left to the reade*
*r.
3.4. Proof of Theorem 3.8 for M = Rk. Analysis of CT(Rk) parallels that of Cn{R*
*k}. A key construc-
tion is that of a map T : NT ! An[Rk], where NT DT(Rk) x [0, 1)V i(T)is a ch*
*osen neighborhood of
DT(Rk) x (tv = 0). Though as mentioned before, DT(Rk) is a subspace of (Rk)E(T)*
*, we emphasize the
role of the vertices of T in the definition of DT(Rk) by naming coordinates on *
*x 2 DT(Rk) as x = (xve),
where v 2 V (T) and e 2 e(V ). Recall that for each v 6= v0 we consider eC#v(Rk*
*) as a subspace of C#v(Rk)
in order to fix each xveas an element of Rk.
Definition 3.19. (1)Let NT(Rk) be the subset of DT(Rk) x [0, 1)V i(T)of poin*
*ts x x (tv), where x
can be any point in DT(Rk), all tv < r(x), defined by
__r(x)__ = 1_min{d(xv, xv)}, wherev 2 V (T), e, e02.E(v)
(1 - r(x)) 3 e e0
(2)By convention, set tv0= 1. Let sw : NT ! [0, 1) send x x (tv) to the prod*
*uct of tv for v in the
root path of w.
(3)For any vertex v of an f-tree T define yv:NT(Rk) ! Rk inductively by sett*
*ing yv0 = 0 and
yv(x) = swxwe+ yw(x), where e is the edge for which v is terminal and w i*
*s the initial vertex of e.
DefinePjTS:fNTr!a(Rk)l(T)togber(y`)l(T).eplacements_
yv0
Figurey3.20.jTvof1the point from Figure 3.7 (and some tv > 0)
yv2 with all yv indicated.
yv3
y1
y4
y7
y6
y3
y2
y5
See Figure 3.20 for an illustration of this construction. The most basic case*
* is when T = the terminal
object of n_, in which case N (Rk) = D (Rk) = Cn(Rk) and j is the canonical i*
*nclusion in (Rk)n_.
Definition 3.21. (1)Given a vertex w of T, let Tw be the f-tree consisting o*
*f all vertices and edges
over w, where w serves as the root of Tw and the leaves over w are re-lab*
*elled consistent with the
order of their labels in T.
(2)Let æw :NT(Rk) ! NTw(Rk) be the projection onto factors indexed by vertic*
*es in Tw, but with tw
set to one and the projection onto eC#w(Rk) composed with the canonical i*
*nclusion to C#w (Rk).
(3)Let fijbe the composite ßi0j0O jTvO æv, where v is the join of the leaves*
* labelled i and j, and
i0 and j0 are the labels of the corresponding leaves of Tv. Similarly, l*
*et gijkbe the composite
si0j0k0O jTw O æw where w is the join of leaves i, j and k.
(4)Define T : NT ! An[Rk] to be the product jT x (fij) x (gijk). Let 0Tbe *
*the restriction of T to
DT x (0)n N.
Proposition 3.22. The image of DT(Rk) under 0Tlies in CT(Rk).
Proof.First note that jT|(ti>0)has image in Cn(Rk). Moreover, if all ti> 0, fij*
*coincides with ßijO jT
and similarly gijk= sijkO jT. Thus, the image of T|(ti>0)lies in ffn(Cn(M)), w*
*hich implies that all of
the image of T, and in particular that of 0T, lies in Cn[M].
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 11
If v is the join of leaves i, j and k and we set (yi) = 0TvO æv(x, (tv)) the*
*n yi0= yj0and thus dijk=
sijk((yi)) = 0 if and only if the join of leaves i and j is some vertex which l*
*ies (strictly) over v. Thus, the
exclusion relation for 0T(x, (tv)) as an element of Cn[M] is the exclusion rel*
*ation associated to T.
The simplest way to see that 0Tis a homeomorphism onto CT(Rk) is to decompos*
*e it as a product and
use our analysis of Cn{Rk} to help define an inverse.
Definition 3.23. Let AT[M] An[Rk] be the subset of points (xi) x (uij) x (dij*
*k) such that
oIf the join of leaves i and j is not the root vertex then xi= xj.
oIf (i, j), k is in the exclusion relation Ex(T) then dijk= 0, dikj= 1, dk*
*ij= 1, etc. and uik= ujk.
Let AT{M} be the subspace of AT[M] for which if there are no exclusions among i*
*, j and k then dijkis
non-zero and finite.
We claim that CT(Rk) = Cn[Rk] \ AT{Rk}. The relations between the xi, uijand *
*dijkwhich hold on
Cn(Rk) also hold on CT(Rk) by continuity. Therefore, the defining conditions of*
* AT{Rk} when restricted
to its intersection with Cn[Rk] will follow from the conditions dijk= 0 when (i*
*, j)k 2 Ex(T), which in turn
are the only defining conditions for CT[Rk].
Thus the image of 0Tlies in AT{Rk}. By accounting for diagonal subspaces and*
* reordering terms, we
will decompose 0Tis a product of maps in order to define its inverse. We first*
* set some notation.
Definition 3.24. Given a map of sets oe : R ! S let pXoe, or just poe, denote t*
*he map from XS to XR
which sends (xi)i2Sto (xoe(j))j2R.
Definition 3.25. (1)Given a tree T choose oe0 : #v0_! n_to be an inclusion o*
*f sets such that each
point in the image labels a leaf which lies over a distinct root edge of *
*T.
(2)Similarly, choose oev : #v_! n_to be an inclusion whose image labels leav*
*es which lie over distinct
edges for which v is initial.
(3)Let pv0: An[Rk] ! A#v0[Rk] be the projection poe0x pC2(oe0)x pC3(oe0).
(4)Similarly, let pv : An[Rk] ! eA#v[Rk] be the projection * x pC2(oev)x pC3*
*(oev).
(5)Let pT be the product (pv)V (T).
For example, with T as in Figure 2.2, the image of oe0 could be {5, 7, 4} and*
* of oev1could be {2, 5}.
Proposition 3.26. For any choices of oev, the projection pT restricted to AT[Rk*
*] is a diffeomorphism onto
i Vji(T)
A#v0[Rk] x eA#v(Rk) , splitting the inclusion of AT[Rk] in An[Rk]. Moreover,*
* composed with this
diffeomorphism, 0Tis the product ff#v0x ('#v). Analogous results hold for AT{R*
*k}.
We leave the proof of this proposition, which is essentially unraveling defin*
*tions, to the reader. We will
now define the inverse to 0Tone vertex at a time. For v 2 V i(T), consder pv(y*
*) 2 eA#v{Rk}, which by
Lemma 1.5 lies in the closure of the image of pv|Cn(Rk). The image of pv|Cn(Rk)*
*)coincides with the image of
'#v, and by Lemma 3.17 the image of '#v is already closed in eA#v{Rk}. Moreover*
*, '#v is a diffeomorphism
onto its image, so we may define the following.
Definition 3.27. (1)For v 2 V i(T), let OEv:CT(Rk) ! eC#v(Rk) = '-1#vO pv.
(2)For v0, note that if y 2 CT(Rk) then pv0(y) lies in the image of ff#v0. D*
*efine OEv0= ff-1#v0O pv0.
V i(T)
(3)Let OET = (OEv)v2V (T): CT(Rk) ! C#v0(Rk) x C#v(Rk) .
12 DEV P. SINHA
In other words, OET is the composite
i jV i(T)
(1) CT(Rk) AT{Rk} pv!A#v0{Rk} x eA#v{Rk}
ff-1#v0x('-1#v)Vi(T)) i Vji(T)
-! C#v0(Rk) x eC#v(Rk) = DT(Rk*
*).
Proof of Theorem 3.8 for M =BRk.y Proposition 3.22, 0Tsends DT(Rk) to CT(Rk) *
* An[Rk]. Defi-
nition 3.27 constructs OET : CT(Rk) ! DT(Rk). By construction, and appeal to Le*
*mma 3.17, they are
inverse to one another. We also need to check that 0Tand (OEv) are smooth, whi*
*ch follows by checking
that their component functions only involve addition, projection and '-1nwhich *
*we know is smooth from
Lemma 3.17.
3.5. Proof of Theorem 3.8 for general M. To establish Theorem 3.8 for general M*
* we first identify
DT(M) as a subspace of DT(Rk), and then we will make use of the established dif*
*feomorphism between
DT(Rk) and CT(Rk). To set notation, let ffl be the given embeddding of M in Rk.
Proposition 3.28. The subspace IDn(M) of M x eCn(Rk) consisting of all (m, x) s*
*uch that all ßij(x) are
in Tm M Rk is diffeomorphic, as a bundle over M, to ICn(M). Through these dif*
*feomorphisms, DT(M)
is a subspace of DT(Rk).
Proof sketch.The first statement follows from the standard identification of TM*
* as a sub-bundle of TRk|M .
The second statement follows from the first statement and Definition 3.6 of D*
*T(M).
From now on, we identify DT(M) with this subspace of DT(Rk).
Proposition 3.29. CT(M) 0T(DT(M)).
Proof.Since CT(M) is already a subspace of CT[Rk], we just need to check that i*
*ts points satisfy the
condition of Proposition 3.28. Looking at (xi) x (uij) x (dijk) 2 Cn(M) inside *
*An[M] we see that the uij
are vectors which are secant to M. Thus, in the closure, if xi= xj then uijis t*
*angent to M at xi.
To prove the converse to this proposition, we show that 0T(DT(M)) lies in CT*
*(M) by modifying the
maps jT and T so that the image of the latter is in the image of ffn. The easi*
*ly remedied defect of jT is
that it maps to the tangent bundle of M in Rk, not to M itself.
Definition 3.30. oLet NT(M) NT(Rk) be the subspace of (x, (tv)) with x 2 *
*DT(M).
oLet j*T: NT(M) ! (Rk)n x (Rk)n = T(Rk)n send (x, (tv)) to jT(x, (0)) x jT*
*(x, (tv)).
Proposition 3.31. The image of j*Tlies in Tffln : TMn T(Rk)n.
Proof.jT(x, (tv)) is defined by adding vectors which by Proposition 3.28 are ta*
*ngent to M to the coordi-
nates of jT(x, (tv)), which are in M.
We map to Mn_by composing with the exponential map Exp(M). For each x 2 DT(M)*
* let Ux be a
neighborhood of x x 0 in NT(M) such that the exponential map Exp(Mn_) is inject*
*ive on j*T(Ux).
Definition 3.32. oLet jM,xT:Ux ! Mn_be the composite Exp(Mn_) O (Tffln)-1O *
*j*T.
oDefine fM,xijby letting (zi) denote jM,xT(y, (tv)) and setting fM,xij= ßi*
*jO jM,xTif zi 6= zj or
DExp O fij, where DExp is the derivative of the exponential map at zi2 TM*
* and the composite
is well defined since TM TRk.
oDefine M,xT: Ux ! An[M] as the product jM,xTx (fMij) x (sijkO jM,xT).
By construction, M,xT|(ti>0)has image in which lies in the image of ffn in A*
*n[M]. On DT(M) \ Ux the
map M,xTcoincides with 0Testablishing that 0T(DT(M)) CT(M). Along with Pro*
*position 3.29 and
the fact that 0Tand its inverse are smooth, this completes the proof of Theore*
*m 3.8.
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 13
4.First properties
Having proven Theorem 3.8 we derive some first consequences from both the the*
*orem itself and the
arguments which went into its proof.
4.1. Characterization in An[M] and standard projections. To map from Cn[M] as w*
*e have defined
it one may simply restrict maps from An[M]. To map into Cn[M] is more difficul*
*t, but the following
theorem gives conditions to verify that some point in An[M] lies in Cn[M].
Theorem 4.1. Cn[Rk] is the subspace of An[Rk] of points (xi) x (uij) x (dijk) s*
*uch that
(1)If xi6= xk then uij= _xi-xk_||xi-xk||and dijk= d(xi,xj)_d(xi,xk).
q _________ 2
(2)If uij, ujk, and uikare all distinct then dijk= 1-(uki.ukj)_1-(uji.u*
*jk)2. Otherwise, if uik= ujk6= uij
then dijk= 0.
(3)uij= -uji, and uij, uikand ujkare linearly dependent Sk-1 with uijbetween*
* ukjand uikon a
great circle in Sk-1.
(4)dijkdikj, dijkdikldiljand dijkdjkidkijare all equal to one.
Moreover, Cn[M] is the subspace of Cn[Rk] where all xi2 M and if xi= xj then ui*
*jis tangent to M at
xi.
Proof.It is simple to check that Cn[Rk] satisfies all of the properties listed.*
* In most cases, the properties
are given by equalities which hold on Cn(Rk) and thus Cn[Rk] by continuity. We *
*noted in Proposition 3.29
that if xi= xj in Cn[M] then uijis tangent to M.
Conversely, we can start with a point x which satisfies these properties, and*
* condition 4 allows us to
define T(x) as in Definition 3.1. We can then either mimic the construction of *
*jT(x)to find points in the
image of ffn nearby showing that x 2 Cn[M], or go through the arguments of sect*
*ion 3 to find an element
of DT(M) which maps to x under T0. The latter argument proceeds by showing tha*
*t x lies in AT(x)[Rk],
as we may use the contrapositives to conditions 1 and 2 along with 3 to show sh*
*ow that if dijk= 0 then
xi= xj and uik= ujk. Then, pv0(x) lies in the image of ff#v0essentially by cond*
*ition 1. Next, pv(x) lies
in the image of '#v by conditions 2, 3 and 4, as these conditions coincide with*
* those given for the image
of ' in Lemma 3.17. We apply the product (OEv) to get a point in DT(Rk) which m*
*aps to x under 0T.
We next turn our attention to the standard projection maps.
Theorem 4.2. By restricting the projection of An[M] onto (M)n to Cn[M], we obta*
*in a projection map
p which is onto, which extends the inclusion ' of Cn(M) in (M)n and for which e*
*very point in Cn(M) has
only one pre-image.
Proof.The fact that p is onto can be seen through composing p with the maps 0T*
*. It is immediate from
definitions that p extends '. Finally, by our characterization in Theorem 4.1, *
*in particular condition 1, any
point in Cn[M] which projects to Cn(M) will be in the image of ffn.
When M = Rk it is meaningful to project onto other factors of An[Rk] to get s*
*imilar extensions.
Theorem 4.3. The maps ßijand sijkextend to maps from Cn[Rk]. Moreover, the exte*
*nsion ßijis an
open map.
Proof.The only statement which is not immediate is that ßijis an open map. We c*
*heck this on each
stratum, using the identification of CT(Rk) as a product to see that when ßijis*
* restricted to CT(Rk) it
factors as pv, where v is the join of leaves i and j, composed with some eßi0j0*
*, each of which is an open
map.
14 DEV P. SINHA
4.2. Manifold structure, codimensions of strata, functoriality for embeddings, *
*and equivari-
ance.
Theorem 4.4. Cn[M] is a manifold with corners for which the M,xTmay serve as c*
*harts.
Proof.The domains of M,xTare manifolds with corners, so it suffices to check t*
*hat these maps are diffeo-
morphisms onto their images in An[M], which is itself a manifold with corners. *
*We have already noted
that M,xTare smooth on their domain, as they are defined using addition in Rk,*
* projection maps, and the
exponential map. Moreover, they may be extended using the same formulas to valu*
*es of tv < 0, as needed
for smoothness with corners.
For M = Rk, the inverse to T is relatively straightforward to define. Given *
*x = (xi) x (uij) x (dijk) 2
An[Rk], first recursively set yv to be the average of yw, where w are terminal *
*vertices for edges coincident
at v, starting with yl= xiwhen l is the leaf labelled by i. We let (yv), as v r*
*anges over terminal vertices
for root edges of T, define xv02 C#v0(Rk).
Along the same lines, for each vertex v first define a point in (Rk)l(Tv)=Sim*
*k by, as in the defintion of
'-1n, setting some xi= 0, some xk = uikand the rest of the xj as dijkxij. Recur*
*sively set xw to be the
average of xu for u directly over w (which is well-defined up to translation an*
*d scaling) and let (xw) as w
ranges over vertices directly over v define xv 2 eC#v(Rk).
Finally, to compute tw we look within the construction above of xv for the ve*
*rtex v over which w sits
directly. Let dw be the greatest distance from one of the of xu, for u over w, *
*to xw, and define dv similarly.
Set tw to be dw=dv.
The map which sends x as above to (xv)x(tv) is the inverse to T, and it is s*
*mooth, defined by averaging
and greatest distance functions. The construction for general M works similarly*
*, by first composing with
the inverse to the exponential map. We leave its construction to the reader.
Since a manifold with corners is a topological manifold with boundary, and a *
*topological with boundary
are homotopy equivalent to their interiors, we get the following.
Corollary 4.5. The inclusion of Cn(M) into Cn[M] is a homotopy equivlance.
An essential piece of data for a manifold with corners are the dimensions of *
*the strata. Dimension
counting for DT(M) leads to the following.
Proposition 4.6. The codimension of CT(M) is #V i(T).
Contrast this with the image of the projection of CT(M) in Mn_, which has cod*
*imension equal to
k . dim(M), where k is the sum over all root edges e of ne- 1 where ne is the n*
*umber of leaves over e.
Next, we have the following long-promised result.
Theorem 4.7. Up to diffeomorphism, Cn[M] is independent of the embedding of M i*
*n Rk.
Proof.The definitions of DT(M) and NT(M) and do not use the embedding of M in R*
*k. Let f and g
be two embeddings of M in Rk, and let f,x_Tand g,x_Tdenote the respective ver*
*sions of M,x_T. Then the
f,x_TO ( g,x_T)-1 compatibly define a diffeomorphism between the two versions *
*of Cn[M].
In fact, since the exponential maps from T(M)n to (M)n are independent of emb*
*edding, so are M,x_T.
Thus, we could use the M,x_Tto topologize the union of the CT(M) without refer*
*ence to An[M]. Yet
another approach would be to first develop Cn[Rk] and then use a diffeomorphism*
* result Theorem 4.7 to
patch Cn[M] together from Cm [Ui] for m n, where Uiis a system of charts for *
*M.
Corollary 4.8. Cn[-] is functorial in that an embedding f :M ! N induces an emb*
*edding of manifolds
with corners Cn[f]: Cn[M] ! Cn[N] which respects the stratifications. Moreover,*
* CT(M) is mapped to
CT(N) by Df on each factor of ICi(M).
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 15
Proof.Since we are free to choose the embedding of M in Rk to define Cn[M] we m*
*ay simply compose the
chosen embedding of N in Rk with f, giving immediately that Cn[M] is a subspace*
* of Cn[N]. Moreover,
by definition of the stratification according to conditions of dijk= 0, CT(M) i*
*s a subspace of CT(M). The
fact that Cn[M] is embedded as a submanifold with corners is readily checked on*
* each stratum, using the
fact that ICi(M) is a submanifold of ICi(N) through Df.
An alternate notation for Cn[f] is evn(f) as it extends the evaluation maps o*
*n Cn(M) and Mn_.
Corollary 4.9. The group of diffeomorphisms of M acts on Cn[M], extending and l*
*ifting its actions on
Cn(M) and Mn_.
The construction of Cn[M] is also compatible with the free symmertric group a*
*ction Cn(M).
Theorem 4.10. The n action on Cn(M) extends to one on Cn[M], which is free and*
* permutes the strata
by diffeomorphisms according to the n action on n_. Thus, the quotient Cn[M]=*
* n is itself a manifold
with corners whose category of strata is isomorphic to e n, the category of unl*
*abelled f-trees.
Proof.The n action on Cn[M] may in fact be defined as the restriction of the a*
*ction on An[M] given by
permutation of indices.
The fact that this action is free follows either from a stratum-by-stratum an*
*alysis or, more directly, from
the fact that if oe is a permutation with a cycle (i1, . .,.ik) with k > 1 and *
*if ui1i2= ui2i3= . .=.uik-1ik= u
then ui1ik= u as well by condition 3 of Theorem 4.1. This implies that uiki1= -*
*u 6= u, which means that
oe cannot fix a point in Cn[M] unless it is the identity.
Finally, note that the coordinate charts M,x_Tcommute with permutation of in*
*dices, so that oeCT(M) =
CoeT(M) through a diffeomorphism, giving rise to a manifold structure on the qu*
*otient.
4.3. The closures of strata. We will now see that the passage from the stratum *
*CT(M) to its closure,
which by Theorem 3.4 consist of the union of CS(M) for S with a morphism to T, *
*is similar to the
construction of Cn[M] itself.
Definition 4.11. Let eCn[Rk] be defined as the closure of eCn(Rk) in eAn[Rk].
Because eAn[Rk] is compact, so is eCn[Rk]. We give an alternate construction *
*of this space as follows.
Extend the action of Simk on (Rk)n_to An[Rk] by acting trivially on the factors*
* of Sk-1and I. This action
preserves the image of ffn and so passes to an action on Cn[Rk]. This is a spec*
*ial case of Corollary 4.9.
Let An[Rk]= ~ and Cn[Rk]= ~ denote the quotients by these actions.
Lemma 4.12. eCn[Rk] is diffeomorphic to Cn[Rk]= ~.
Proof.First note that An[Rk]= ~ is compact, and thus so is Cn[Rk]= ~. The proj*
*ection map from
An[Rk]= ~ to eAn[Rk] thus sends sends Cn[Rk]= ~ onto eCn[Rk] by Lemma 1.5. In t*
*he other direction, we
may essentially use the maps '-1kto define an inverse to this projection, by re*
*constructing a point in (Rk)n_
up to translation and scaling from its images under ßijand sijk.
We may define a stratification of eCn[Rk] labelled by trees in the same fashi*
*on as for Cn[Rk], and the
strata have a more uniform description than that of CT(Rk).
i jV (T)
Corollary 4.13. eCT(Rk) is diffeomorphic to eC#v(Rk) .
Proof.We cite Lemma 4.12 and check that Simk is acting on each CT(Rk) non-trivi*
*ally only on the factor
of C#v0(Rk), and doing so there by its standard diagonal action.
Other results for Cn[Rk] have similar analogues for eCn[Rk], which we will no*
*t state in general. One of
note is that its category of strata is isomorphic to e n, the category of trees*
* with a trunk.
16 DEV P. SINHA
Definition 4.14. (1)Define ICn[M] as a fiber bundle over M with fiber eCn[Rm*
* ] built from TM by
taking the same system of charts but choosing coordinate transformations *
*eCn[OEij] from eCn[Rm ] to
itself, where OEijare the coordinate transformations defining TM.
(2)Let ICe[M] be defined as in Definition 3.6 but with ICn[M] replacing ICn(*
*M).
(3)Let DT[M] be defined through the pull-back
DT[M] ----! (ICe[M])E0
?? ?
y ?y
C#v0[M] ----! (M)E0.
Theorem 4.15. CT[M] is diffeomorphic to DT[M].
Proof.Though by definition CT[M] is the closure of CT(M) in Cn[M], it is also t*
*he closure of CT(M)
in any closed subspace of An[M], and we choose to consider it as a subspace of *
*AT[M]. The inclusion of
CT(M) in AT[M] is compatible with fiber bundle structures of these spaces over *
*(M)E0. For a general fiber
bundle F0! E0! B0subspaces respectively of F ! E ! B, the closure clE(E0) may b*
*e defined by first
extending E0to a bundle over clB(B0) (which may be done locally) and then takin*
*g the closures fiber-wise.
Our result follows from this general statement, the definition of C#v0[M] as th*
*e closure of C#v0(M) in
A#v0[M], and the independence of the closure of the fibers eCi(Rm ) of ICi(M) i*
*n any eAi[Rk].
4.4. Configurations in the line and associahedra. The compactification of confi*
*gurations of points
in the line is a fundamental case of this construction. The configuration spac*
*es Cn(R) and Cn(I) are
disconnected, having one component for each ordering of n points. These differe*
*nt components each map
to a different component of An[R], because whether xi < xj or xi > xj will dete*
*rmine a + or - for
uij2 S0. Let Con[R] and Con[I] denote the closure of the single component x1 < *
*. .<.xn.
The main result of this subsection is that eCn[R] is Stasheff's associahedron*
* An-2, of which there is
a pleasing description of An due to Devadoss. The truncation of a polyhedron a*
*t some face (of any
codimension) is the polyhedral subspace of points which are of a distance great*
*er than some sufficiently
small epsilon from that face. We may define An as a truncation of n. In the st*
*andard way, label the
codimension one faces of n with elements of n_+_1_. Call S n_+_1_consecutive*
* if i, j 2 S and i < k < j
implies k 2 S, and call a face of n consecutive if the labels of codimension o*
*ne faces containing it are
consecutive. To obtain An, truncate the consecutive faces of n, starting with *
*the vertices, then the edges,
and so forth.
Figure 4.16.The third associahedron.
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 17
We will use a more conventional definition of the associahedron below. Closel*
*y related to the associa-
hedron is the following sub-category of n, whose minimal objects correspond to*
* ways in which one can
associate a product of n factors in a given order.
Definition 4.17. Let ondenote the full sub-category of n whose objects are f-*
*trees such that the set
of leaves over any vertex is consecutive and such that the root vertex has vale*
*nce greater than one.
Note that any element of onhas an embedding in the upper half plane with the*
* root at 0, in which
the leaves occur in order and which is unique up to isotopy. We may then drop t*
*he labels from such an
embedding.
In applications to knot theory, we consider manifolds with boundary which hav*
*e two distinguished
points in its boundary, the interval I being a fundamental case.
Definition 4.18. Given such a manifold M with y0 and y1 in @M, let Cn[M, @] be *
*the subpsace of
Cn+2[M] which is the preimage under p of points of the form (y0, x1, . .,.xn, y*
*1) 2 (M)n+2.
Theorem 4.19. Stasheff's associahedron An, eCon+2[R] and Con[I, @] are all diff*
*eomorphic as manifolds with
corners. Moreover, their barycentric subdivisions are diffeomorphic to the real*
*ization (or order complex)
of the poset on.
Proof.It is simple to check that eCon+2[R] and Con[I, @] are diffeomorphic usin*
*g Lemma 4.12 and the fact
that up to translation and scaling any x0 < x1 < . .<.xn+1 R has x0 = 0 and xn*
*+1= 1.
Next, we analyze eCon+2[R] inductively using Corollary 4.13 and Theorem 4.15.*
* First note that because
the xiare ordered and x0 can never equal xn+1, the category of strata of eCon+2*
*[R] is on+2. For n + 2 = 3,
Ceo3[R] is a one-manifold whose interior is the open interval eCo3(R) and which*
* according to o3has two
distinct boundary points, and thus must be an interval. For n + 2 = 4, the stra*
*tification according to
o4and Theorem 4.15 dictate that there are five codimension one boundary strata*
* each isomorphic to
Ceo3[R], which we know inductively to be I, and five vertices, each being the b*
*oundary of exactly two faces,
attached smoothly (with corners) to an open two-disk, making a pentagon.
In general, eCon+2[R] has an open n-ball for an interior and faces eCoT[R] wh*
*ich inductively we identify
as (A#v-2)v2V (T)glued according to the poset structure of on+2to make a bound*
*ary sphere, coinciding
with a standard definition of An using trees [23].
The last statement of the theorem follows from the general fact that if P is *
*a polytope each of whose
faces (including itself) is homeomorphic to a disk, then the realization of the*
* category of strata of P is
diffeomorphic to its barycentric subdivision.
In further work [18] we plan to show that the spaces eCn[Rk] form an operad. *
*This construction unifies
the associahedra and little disks operads, and was first noticed in [12].
To review some of the salient features of the structure of Cn[M] in general, *
*it is helpful to think explictly
about coordinates on Co2[I, @]. On its interior, suitable coordinates are 0 < x*
* < y < 1. Three of the faces
are standard, corresponding to those for 2. They are naturally labelled x = 0,*
* y = 1 and x = y, and for
example we may use y as a coordinate on the x = 0 face, extending the coordinat*
*es on the interior. The
final two faces are naturally labelled 0 = x = y and x = y = 1. Coordinates on *
*these faces which extend
interior coordinates would be x_yand 1-y_1-x, respectively.
18 DEV P. SINHA
Figure 4.20.The second associahedron, labelled by 04, with
labellings by associativity and coordinates also indicated.
5.The simplicial compactification
Recall Definition 1.3 of Cn<[M]>, which we call the simplicial compactificati*
*on. For M = I, we see that
Con<[I]> is the closure of Con(I) in In_, which is simply n. For general manif*
*olds, we will see that Cn<[M]>
is in some sense more complicated than Cn[M].
Because the projection PA from An[M] onto An<[M]> commutes with the inclusion*
*s of Cn(M), Lemma 1.5
says that PA sends Cn[M] onto Cn<[M]> when M is compact, as we assume throughou*
*t this section.
Definition 5.1. Let Qn:Cn[M] ! Cn<[M]> be the restriction of PA.
The aim of this section is to understand Qn, and thus the topology of Cn<[M]>*
*, in particular showing
that this projection map is a homotopy equivalence. An immediate consequence of*
* the surjectivity of Qn
and Theorem 4.1 is the following.
Theorem 5.2. Cn<[Rk]> is the subspace of An<[Rk]> of points (xi) x (uij) such t*
*hat
(1)If xi6= xk then uij= _xi-xk_||xi-xk||.
(2)uij= -uji, and uij, uikand ujk are linearly dependent with uijbetween ukj*
* and uikon the
geodesic between them in Sk-1.
Moreover, Cn<[M]> is the subspace of Cn<[Rk]> where all xi2 M and if xi= xj the*
*n uijis tangent to M
at xi.
From the analysis of Lemma 3.17 we know that (ßij) : eCn(Rk) ! (Sk-1)n(n-1)is*
* not injective for
configurations in which all points lie on a line. These collinear configuratio*
*ns account for all of the
difference between Cn[M] and Cn<[M]>.
Lemma 5.3. The map Qn is one-to-one except at points with some xi1= . .=.xim an*
*d uihij= uiki`
for any h, j, k, `. The preimages of such points are diffeomorphic to a product*
* of Am-2's.
Proof.Conditions 1 and 2 of Theorem 4.1 say that in cases except these, the coo*
*rdinates dijkin Cn[M]
will be determined by the xior uijcoordinates. In these cases, the dijkare rest*
*ricted in precisely the same
manner as for the definition of eCom(R), which is diffeomorphic to Am-2 by Theo*
*rem 4.19.
Thus, the preimage of any point under Qn will be contractible, pointing to th*
*e fact that Qn is a homotopy
equivalence. A small difficulty is that under Qn points in the boundary of CT[M*
*] will be identified with
points in its interior. Moreover, there are identifications made which lie only*
* in the boundary of CT[M].
We will first treat configurations in Rk up to the action of Simk, the building*
* blocks for the strata of
Cn[M].
Definition 5.4. Let eCn<[Rk]> be the closure of the image of eCn(Rk) under (eßi*
*j) to eAn<[Rk]> = (Sk-1)C2(n_).
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 19
The analogue of Lemma 4.12 does not hold in this setting, since as noted befo*
*re (eßij) is not injective for
collinear configurations. Nonetheless, we will see that eQn: eCn[Rk] ! eCn<[Rk]*
*> is a homotopy equivalence
by exhibiting eQnas a push-out by an equivalence. We first state some generalit*
*ies about fat wedges and
pushouts.
Definition 5.5. Let {Ai Xi} be a collection of subpsaces indexed by i in some *
*finite I. Define the fat
wedge of {Xi} at {Ai}, denoted IAiXior just AiXi, to be the subspace of (xi) *
*2 (Xi)I with at least
one xiin Ai.
Suppose for each i we have a map qi: Ai! Biand let Yibe defined by the follow*
*ing push-out square
Ai ----! Xi
? ?
qi?y ~qi?y
Bi ----! Yi.
There is a map which we call qifrom AiXito BiYi.
Lemma 5.6. With notation as above, if each Ai ,! Xi is a (Hurewicz) cofibration*
* and each qi is a
homotopy equivalence then qiis a homotopy equivalence.
Proof.First note that in a left proper model category, if you have a diagram
B ---- A ----! C
?? ? ?
y ?y ?y
X ---- Y ----! Z,
where the vertical maps are equivalences and at least one map on each of the ho*
*rizontal levels is a cofi-
bration, then the induced map of pushouts is an equivalence (see Theorem 13.5.4*
* in [13]). The Hurewicz
model category is left proper because every space is cofibrant (see Theorem 13.*
*1.3 in [13]).
We prove this lemma by induction. Let I = {1, . .,.n}. Inductively define the*
* diagram Dj as
Pj-1x Xj ---- Pj-1x Aj ----! (Xi)i, factoring eQn. We will see that this map is a*
* homeomorphism on the
image of eCm(Rk) in Rm , but not on its boundary strata.
*
* i Vj(T)
Definition 5.8. oBy the analogue of Theorem 4.15, eCT[Rk] is diffeomorphic *
*to gC#v[Rk] .
Let ~T CT[Rk] be the fat wedge ~#vgC#v[Rk].
oLet LT denote the fat wedge Sk-1x #vR#v and let qT = q#v : ~T ! LT.
S S
oLet T~T denote the union of the ~T in eCn[Rk]. Let TLT denoteSthe un*
*ionSof LTSwith
identifications qT(x) ~ qT0(y) if x 2 ~T is equal to y 2 ~T0. Let T qT :*
* T ~T ! T LT denote
the projection defined compatibly by the qT.
Theorem 5.9. The projection map eQn: eCn[Rk] ! eCn<[Rk]> sits in a pushout squa*
*re
S e k
T?~T ----! Cn[R?]
(3) STqT?y Qn?y
S e k
TLT ----! Cn<[R ]>.
Before proving this theorem we deduce from it one of the main results of this*
* section.
Corollary 5.10. eQnis a homotopy equivalence.
Proof.If we apply Lemma 5.6 to the push-out squares of Equation 2 whichSdefine *
*the R#v, we deduce
thatSqT is a homotopy equivalence.SBecause of the identifications in T LT areS*
*essentially defined through
TqT, we deduce that T qT is a homotopy equivalence. Because the inclusion T*
* ~T ! eCn[Rk] is a
cofibration, we see that eQnis a pushout of a homotopy equivalence through a co*
*fibration, and thus is a
homotopy equivalence itself.
Proof of Theorem 5.9.Let X denoteSthe pushoutSof the first three spaces in the *
*square of EquationS3. First
note that the composite eQnO ( TqT)-1 : T LT ! eCn<[Rk]> is well-defined, sin*
*ce choices of ( TqT)-1
only differ in their dijkcoordinates. By the definition of push-out, X maps to *
*eCn<[Rk]> compatibly with
Qen. We show that this map F is a homeomorphism.
*
* S
First, F is onto because eQnis onto. The key is that by construction F is one*
*-to-one. Away from T ~T,
Qenis one-to-one essentially by Lemma 5.3. The projection eQnis not one-to-one *
*only on x 2 eCm[Rk] with
*
* S
some collections of {ij} such that uihij=S ui`im. But such an x is in ~T(x). Th*
*e map eQnO ( TqT)-1S
is one-to-one since distinct points in T LT will have distinct uijcoordinates *
*when lifted to T ~T which
remain distinct in eCn<[Rk]>.
Finally since it is a push-out of compact spaces, X is compact. All spaces in*
* question are subspaces of
metric spaces. Thus, since F is a one-to-one map between metrizable spaces whos*
*e domain is compact, it
is a homeomorphism onto its image, which is all of eCn<[Rk]>.
Theorem 5.11. The map Qn:Cn[M] ! Cn<[M]> is a homotopy equivalence.
Proof.On the interior Cn(M), Qn is a homeomorphism.
The effect of Qn on CT[M] for non-trivial T is through restriction to P#v0 on*
* the base C#v0[M].
Working fiberwise, we see Qn takes each fiber bundle eCi[Rm ] ! ICi[M] ! M and *
*pushes out fiberwise to
get eCi<[Rm ]> ! ICi<[M]> ! M. As #v0 < n, by induction and Theorem 5.9, Qn res*
*tricted to any CT[M]
is a homotopy equivalence. Since the inlcusions of CT[M] in each other are cofi*
*brations, we can build a
homotopy inverse inductively and deduce that Qn is a homotopy equivalence.
Unfortunately, Cn<[M]> is not a manifoldSwith corners. It is however stratif*
*ied by manifolds.S For
example, eCn<[Rk]> is the union of T LT, a union of manifolds, and the complem*
*ent of T ~T which is a
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 21
submanifold of eCn[Rk]. The singularity which arises is akin to that which occu*
*rs when say a diameter of
a disk gets identified to a point. We will not pursue the matter further here.
6.Diagonal and projection maps
As we have seen, the compactifications Cn[M] and Cn<[M]> are functorial with *
*respect to embeddings
of M. In this section we deal with projection and diagonal maps, leading to fun*
*ctorality with respect to
n, viewed as the set n_.
Our goal is to construct maps for C#S[M] and C#S<[M]> which lift the canonica*
*l maps on MS. We start
with the straightforward case of projection maps. If oe : m_! n_is an inclusion*
* of sets, recall Definition 3.24
that pMoeis the projection onto coordinates in the image of oe.
Proposition 6.1. Let oe : m_! n_be an inclusion of finite sets. There are proje*
*ctions Coefrom Cn[M] onto
Cm [M] and from Cn<[M]> onto Cm <[M]> which commute with each other, with pMoe,*
* and its restriction to
Cn(M).
Proof.The inclusion oe gives rise to maps from Ci(oe) : Ci(m_) ! Ci(n_). We pro*
*ject An[M] onto Am [M]
through Poe= pMoex pSk-1C2(oe)x pIC3oe.
Because P O ffn = ffm and all spaces in question are compact we apply Lemma 1*
*.5 to see that Poesends
Cn[M] onto Cm [M], extending the projection from Cn(M) to Cm (M). By constructi*
*on, Poecommutes
with with pMoe, which is its first factor.
The projection for Cn<[M]> is entirely analogous, defined as the restriction *
*of the map P0oe= pMoexpSk-1C2(oe):
An<[M]> ! Am <[M]>. We leave the routine verification that P0 commutes with all*
* maps in the statement
of the theorem to the reader.
An inclusion oe : m_! n_gives rise to a functor Exoe: Ex(n_) ! Ex(m_) by thro*
*wing out any exclusions
involving indices not in the image of oe. The corresponding "pruning" functor f*
*or trees, oe: n_! m_, is
defined by removing leaf vertices and edges whose label is not in the image of *
*oe, replacing any non-root
bivalent vertex along with its two edges with a single edge, and removing any v*
*ertices and edges which
have all of the leaves above them removed.
Proposition 6.2. Coesends CT[M] to C oe(T)[M].
Proof.The effect of Coeis to omit indices not in the image of oe, so its effect*
* on exclusion relations is
precisely Exoe. There is a univalent root vertex for the tree associated to Coe*
*(CT[M]) if and only if all
indices j for which xj 6= xihave been omitted, which happens precisely when all*
* leaves in T except for
those over a single root edge have been pruned.
If oe : m_! n_is not injective, it is more problematic to construct a corresp*
*onding map Cn[M] ! Cm [M].
Indeed, poe: Mn_! Mm_will not send Cn(M) to Cm (M), since the image of poewill *
*be some diagonal
subspace of Mm_and the diagonal subspaces are precisely what are removed in def*
*ining Cn(M). One can
attempt to define diagonal maps by öd ubling" points, that is adding a point to*
* a configuration which is
very close to one of the points in the configuration, but such constructions ar*
*e non-canonical and will never
satisfy identities which diagonal maps and projections together usually do. But*
*, the doubling idea carries
through remarkably well for compactified configuration spaces where one can öd *
*uble infinitesimally".
From the viewpoint of applications in algebraic topology, where projection and *
*diagonal maps are used
frequently, the diagonal maps for compactifications of configuration spaces sho*
*uld be of great utility.
Reflecting on the idea of doubling a point in a configuration, we see that do*
*ing so entails choosing
a direction, or a unit tangent vector, at that point. Thus we first incorporat*
*e tangent vectors in our
constructions. Recall that we use STM to denote the unit tangent bundle (that i*
*s, the sphere bundle to
the tangent bundle) of M.
22 DEV P. SINHA
Definition 6.3. If Xn(M) is a space with a canonical map to Mn_, define X0n(M) *
*as a pull-back as follows
X0n(M)----! (STM)n_
?? ?
y ?y
Xn(M) ----! Mn_.
If fn : Xn(M) ! Yn(M) is a map over Mn_, let f0n: X0n(M) ! Yn0(M) be the induce*
*d map on pull-backs.
Lemma 6.4. C0n[M] is the closure of the image of ff0n: C0n(M) ! A0n[M]. Simila*
*rly, C0n<[M]> is the
closure of the image of fi0n.
Proof.We check that clA0n[M](ff0n(C0n(M)))satisfies the definition of C0n[M] as*
* a pull-back by applying
Lemma 1.5 with ß being the projection from A0n[M] to An[M] and A being the subs*
*pace ffn(C0n(M)). The
proof for C0n<[M]> proceeds similarly.
We may now treat both diagonal and projection maps for C0n<[M]>. Starting wit*
*h M = Rk, note that
A0n<[Rk]> = (Rkx Sk-1)n_x (Sk-1)C2(n_), which is canonically diffeomorphic to (*
*Rk)n_x (Sk-1)n_2, as we let
uiibe the unit tangent vector associated to the ith factor of Rk.
Definition 6.5. Using the productkdecompositionkabove-and1considering M as a su*
*bmanifold of Rk, define
Aoe: A0n<[Rk]> ! A0m<[Rk]> as pRoex pSoe2and let Foebe the restriction of Aoeto*
* Cn<[M]>.
Proposition 6.6. Given oe : m_! n_the induced map Foesends C0n<[M]> to C0m<[M]>*
* and commutes with
pSTMoe.
Proof.To see that the image of Foelies in C0m<[M]>, it suffices to perform the *
*routine check that its
projection to Am <[M]> satisfies the conditions of Theorem 5.2 using the fact t*
*hat the domain of Foe,
namely C0n<[M]>, satisfies similar conditions. Let (xi) x (uij) be Foe((y`) x (*
*v`m))so that xi= yoe(i)and
uij= voe(i)oe(j).
Looking at the first condition of Theorem 5.2, xi6= xj means yoe(i)6= yoe(j).*
* By Theorem 5.2 applied
to Cn<[M]> we have that voe(i)oe(j)is the unit vector from yoe(i)to yoe(j), whi*
*ch implies the corresponding
fact for uij. Checking that uij= -ujifor i 6= j is also immediate in this way, *
*as is checking the linear
dependence condition on uij, uikand ujkif i, j and k are distinct. If i, j and *
*k are not distinct, dependence
is even easier to check, since two of these vectors will be equal up to sign.
We leave the rest of these routine checks to the reader.
Let N denote the full subcategory of the category of sets generated by the n_.
Corollary 6.7. Sending n_to C0n<[M]> and oe to Foedefines a contravariant funct*
*or from N to spaces.
Proof.We check that FoeOø= FoeOFø. This follows from checking the analogous fac*
*ts for poeand poe2, which
are immediate.
Let [n] = {0, . .,.n}, an ordered set given the standard ordering of integers*
*. Recall the category
, which has one object for each nonnegative n and whose morphisms are the non-*
*decreasing ordered
set morphisms between the [n]. A functor from to spaces is called a cosimpli*
*cial space. There is a
canonical cosimplicial space often denote o whose nth object is n. To be defi*
*nite we coordinatize n
by 0 = t0 t1 . . .tn tn+1 = 1, and label its vertices by elements of [n] *
*according to the number
of tiequal to one. The structure maps for this standard object are the linear m*
*aps extending the maps of
vertices as sets. On coordinates, the linear map corresponding to some oe : [n]*
* ! [m] sends (ti) 2 n to
(toe*(j)) 2 m where n - oe*(j) is the number of i 2 [n] such that oe(i) < m - *
*j.
The following corollary gives us another reason to refer to Cn<[M]> as the si*
*mplicial compactification
of Cn(M). For applications we are interested in a manifold M equipped with one *
*inward-pointing tangent
vector v0 and one outward-pointing unit tangent vector v1 on its boundary. Let *
*C0n<[M, @]> denote the
MANIFOLD-THEORETIC COMPACTIFICATIONS OF CONFIGURATION SPACES *
* 23
subspace of C0n+2<[M, @]> whose first projection onto STM is v0 and whose n + 2*
*nd projection is v1. Let
OE : ! N be the functor which sends [n] to n_+_1_and relabels the morphism ac*
*cordingly.
Corollary 6.8. The functor which sends [n] to C0n<[M, @]> and oe : [n] ! [m] to*
* the restriction of pø to
C0n<[M, @]> where ø : [m + 1] ! [n + 1] is the composite OE O oe*O OE-1 defines*
* a cosimplicial space.
This cosimplicial space models the space of knots in M [22].
For C0n[M], projection maps still work as in Proposition 6.1, but diagonal ma*
*ps are less canonical and
more involved to described. We restrict to a special class of diagonal maps for*
* simplicity.
Definition 6.9. Let oei: n_+_k_! n_be defined by letting Ki= {i, i + 1, . .,.i *
*+ k} and setting
8
>i j 2 Ki
:j - k j > Ki.
We must take products with associahedra in order to account for all possible *
*diagonal maps.
Definition 6.10. oDefine 'i: IC3(n_)xAk-1! IC3(n+k_)by recalling that Ak-1~*
*=C^k+1(R) IC3(k_)
and sending (dj`m)C3(n_)x (ej`m)C3(k_)to (fj`m)C3(n+k_)with
8
>>>doei(j,`,m)if at most onejof, `, m 2 Ki
>>>
<0 ifj, ` 2 Kibutm =2Ki
fj`m= >1 if`, m 2 Kibutj =2Ki
>>>
>>:1 ifj, m 2 Kibut` =2Ki
ej-i,`-i,m-iifj, `, m 2 Ki.
oLet Di,k: A0n[M] x Ak-1! A0n+k[M] be the product of Aoei: A0n<[M]> ! A0n+*
*k<[M]> with 'i. Let
ffiikdenote the restriction of Dito C0n[M] x Ak-1.
Proposition 6.11. ffiiksends C0n[M] x Ak-1to C0n+k[M] A0n+k[M].
As with Proposition 6.6, the proof is a straightforward checking that the ima*
*ge of ffiiksatisfies the
conditions of Theorem 4.1. One uses the fact that C0n[M] satisfies those condit*
*ions, along with the definition
of 'i. We leave closer analysis to the reader.
By analysis of the exclusion relation, we see that the image of ffiiklies in *
*C0S[M] where S is the tree with
n + k leaves where leaves with labels in Kisit over the lone one interal vertex*
*, which is initial for the ith
root edge. In general, ffiiksends C0T[M] to C0T0[M], where T0 is obtained from *
*T by adding k + 1 leaves to
T, each of which has the ith leaf as its initial vertex.
We set ffii = ffii1: C0n[M] ! C0n+1[M], and note that these act as diagonal m*
*aps. One can check that
composing this with the projection down back to C0n[M] is the identity. Unfortu*
*nately, ffiiffii6= ffii+1ffii- see
Figure 6.12 - so that the C0n[M] do not form a cosimplicial space. But note tha*
*t our ffi2, when we restrict
A1 to its boundary, restricts to these two maps and thus provides a canonical h*
*omotopy between them. In
fact Proposition 6.11 could be used to make an A1 cosimplicial space, but it is*
* simpler to use the C0n<[M]>
if possible.
24 DEV P. SINHA
Figure 6.12.An illustration that ffi2ffi2 6= ffi3ffi2.
PSfrag_replacements_
ffi2
ffi2
ffi3
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Department of Mathematics, University of Oregon, Eugene, OR 97403
E-mail address: dps@math.uoregon.edu