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 References [1]S. Axelrod and I. Singer. Chern-Simons perturbation theory, II. Journal of * *Differential Geometry 39 (1994), no. 1, 173-213. [2]D/ Bar-Natan, S. Garoufalidis, L. Rozansky, and D. Thurston. The rhus integ* *ral of rational homology 3-spheres. I. A highly non trivial flat connection on S3. Selecta Math. (N.S.) 8 (2002), no.* * 3, 315-339. [3]R. Bott and C. Taubes. On the self-linking of knots. Topology and physics. * *J. Math. Phys. 35 (1994), no. 10, 5247-5287. [4]R. Budney, J. Conant, K. Scannell and D. Sinha. New perspectives on self-li* *nking. Submitted, 2003. [5]A. Cattaneo, P. 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