SCATTERING CONFIGURATION SPACES

SCATTERING CONFIGURATION SPACES RICHARD MELROSE AND MICHAEL SINGER 1.0B; Revised: 14-8-2008; Run: March 17, 2009 Astract. For a compact manifold with oundary X we introduce the n-fold scattering stretched product X n which is a compact manifold with corners sc for each n; coinciding with the previously known cases for n = 2; 3: It is constructed y iterated low up of oundary faces and oundary faces of multi-diagonals in X n : The resulting space is shown to map smoothly, y a -ration, covering the usual projection, to the lower stretched products. It is anticipated that this manifold with corners, or at least its cominatorial structure, is a universal model for phenomena on asymptotically at manifolds in which particle clusters emerge at innity. In particular this is the case for magnetic monopoles on R 3 in which case these spaces are closely related to compactications of the moduli spaces with the oundary faces mapping to lower charge idealized moduli spaces. Contents Introduction 1 1. Manifolds with corners 6 2. Boundary low up 8 3. Intersection-orders 11 4. Boundary conguration spaces 16 5. Multi-diagonals 19 6. D-collections 22 7. Boundary diagonals 25 8. Scattering conguration spaces 27 9. Reordering low-ups 27 10. Scattering stretched projections 31 11. Vector spaces 32 References 33 Introduction One of the natural tools for the study of asymptotically translation-invariant phenomena on Euclidean spaces is the passage to the `radial compactication' X = R m = R m [ S1 m 1 : This translates asymptotic ehaviour to ehaviour at the oundary of X and allows similar phenomena to e considered on aritrary compact manifolds with oundary, in terms of the intrinsic scattering structure at the oundary. In this approach, emphasized in [4], typical kernels and functions, such as Euclidean distance, d(z 1 ; z 2 ) 2 = jz 1 z 2 j 2 which are quite singular near the 1

% 2 RICHARD MELROSE AND MICHAEL SINGER corner of the compact space X X are resolved to `normal crossings', i.e. conormal singularities, y lifting to the the scattering stretched product Xsc: 2 This space is otained y iterated real low-up of X 2 : The corresponding triple product Xsc 3 has also een discussed and here we consider the `higher scattering products' of an aritrary compact manifold with oundary. These inherit the permutation invariance of X n and, apart from their construction, the most important result here is that the projections onto smaller products also lift to e smooth -rations, giving (I.1) X n sc X n 1 sc : : : X 4 sc X 3 sc X 2 sc It is our asic contention that the spaces Xsc; n despite the apparent complexity or their denition, are universal for asymptotically translation-invariant phenomena. There is of course a relation etween the spaces considered here and the seminal work of Fulton and MacPherson, [1]. This relationship is strongest at a cominatorial level ut the dierences are also quite sustantial. Apart from the distinction etween real and complex spaces in the two settings, it should e noted that all the low ups carried out here are in the oundary. As a result the space Xsc n can e retracted to X n and in this sense the topology has not changed at all. These spaces are designed to give `more room' for geometric and analytic ojects. One justication for the introduction of these `higher' stretched products is that they are anticipated to serve as at least cominatorial models for a natural compactication of the moduli space of magnetic monopoles on R 3 and allow the detailed asymptotic description of the hyper-kahler metric. This will e shown elsewhere and is closely related to the existence of the maps in (I.1). Given the permutation-invariance of the spaces, the existence of these maps can e reduced to one case in each dimension and then corresponds to a commutative diagramme X: n;sc (I.2) X n sc Xsc n 1 X Xsc n 1 X n n X n 1 where n;sc is the new map and n is projection o the last factor. In fact this lifted map is a -ration, meaning in particular that push-forward under it of a function with complete asymptotic expansion (so in essence `smooth' up to the oundary faces) again has such an expansion. In the rest of this Introduction, we give an outline of this construction, deferring the proofs to the main ody of the article. Given their fundamentally cominatorial nature the constructions here also extend to the more general `red-cusp' conguration spaces Xn ; corresponding to the doule and triple spaces introduced y Mazzeo and the rst author in [2]. These arise naturally when @X comes with a ration over a space Y: The scattering case appears when this ration is the identity map @X! @X: The main issue in what follows is the fundamental fact aout iterated low-ups, which is that dierent orders generally lead to non-dieomorphic spaces. Thus while it is clear which sumanifolds of the oundary must e lown up in the construction of Xsc; n the order in which this is done has to e specied. Moreover, having

SCATTERING CONFIGURATION SPACES 3 determined this order, the existence of the map n;sc in (I.2) corresponds to the possiility of otaining the same space y a performing these low ups in a dierent order. In a manifold with corners, M; such as X n and the manifolds otained y susequent low up from it, admissile `centres' of low up, H M; which is to say manifolds which have a collar neighourhood, are called p-sumanifolds (for product-) and are always required here to e closed. The low up of a p-sumanifold is then always possile and is denoted, with its lown-down map, (I.3) : [M; H]! M: Thus we are interested in the circumstances in which two p-sumanifolds H 1 and H 2 of a manifold with corners commute in the sense that the natural identication of the complement of the preimage of H 1 [ H 2 in the lown up spaces extends to a smooth dieomorphism allowing us to identify (I.4) [M; H 1 ; H 2 ] = [M; H 2 ; H 1 ]: Here, and throughout the paper, we have identied sumanifolds with their lifts to the low-up. The lift of H 2 to [X; H 1 ] is the closure in [X; H 1 ] of 1 1 (H 2 n H 1 ) if this is non-empty and is 1 1 (H 2) in the opposite case (i.e. if H 2 H 1 ). In particular to conclude (I.4) we need to know that each lifts to e a p-sumanifold under the low-up of the other. In fact, as is well-known, H 1 and H 2 commute in this sense if and only if they are either transversal (including the case that they are disjoint) or comparale, meaning either H 1 H 2 or H 2 H 1 : To prove our results, we need to show that whole families of low-ups commute and this is where the cominatorial complexity lies. I.1. Boundary products. The `scattering structure' on a manifold with corners can e identied with the intrinsic Lie algera of `scattering vector elds' consisting precisely of the products fv where V 2 V (X); meaning that V is a smooth vector eld which is tangent to the oundary, and f 2 C 1 (X) vanishes at the oundary. The larger Lie algera V (X) is the `oundary (-) structure'; it can also e thought of as representing the asymptotic multiplicative structure near the oundary or geometrically as the `cylindrical end' structure on the manifold. Not surprisingly then, our construction egins from the corresponding n-fold stretched -product X n : This resolves, near the diagonal in the oundary, the pairwise distance functions (and of course much more esides) for a cylindrical-end metric, also called a -metric, on X: Assuming, for simplicity and from now on, that @X is connected, X n is otained from Xn y the low up of all oundary faces. In order to descrie the construction, consider the collection of all oundary faces, M(X n ); of X n : Blowing up oundary hypersurfaces of a manifold with corners does noting, so let B M(X n ) e the suset of oundary faces of X n of codimension at least 2; if @X is not connected B is a smaller collection. As standard notation we shall not distinguish etween oundary faces of X n and their lifts to lown up versions of the manifold, except where asolutely necessary. Denition I.1. The -stretched products of X; X n (I.5) X n = [X n ; B n ; B n 1 ; : : : ; B 2 ]: are dened to e where B r M(X n ) is the collection of oundary faces of codimension r:

& 4 RICHARD MELROSE AND MICHAEL SINGER Note the contracted notation in (I.5) for iterated low ups. Since we have not specied an ordering of elements within each of the families B r ; it is implicit that the result does not depend on these choices. In fact at the stage at which the elements of B r are lown up they are disjoint so the order is immaterial and X n is well-dened. It also follows from this that the permutation group lifts to X n as dieomorphisms. Consider the analogue of (I.2) in this simpler setting: n; (I.6) X n X n 1 X X n 1 X n n X n 1 To show the existence of n; we divide B r into two pieces, the vertical and nonvertical oundary faces (with respect to the projection n ): Namely Br v consists of those oundary faces B of X n of codimension r such that the n-th factor of B is X: Similarly, Br nv consists of those oundary faces B of X n of codimension r such that the n-th factor of B is @X: Thus B 2 B v if B = B0 X with B 0 2 M(X n 1 ); i.e. B = n (B) X: Otherwise the nth factor is necessarily @X and then B 2 B nv: Equivalently, the vertical oundary faces are those that are unions of res of n : Now oserve that (I.7) X n 1 X = [X n ; B v n 1; : : : ; B v 2] since the last factor of X is unchanged throughout. Thus the existence of n; in (I.6) follows if we show that the non-vertical oundary faces can all e commuted to come last and hence that so exhiiting X n X n = [X n 1 X; B nv n ; B nv n 1; : : : ; B nv 2 ]; is an iterated low-up of Xn 1 X: I.2. Multi-diagonals. The space Xsc n is constructed from X n y the low up of the intersections of the (lifts of the) `multi-diagonals' in X n with the various oundary components of X n ; again with strong restrictions on the order in which this is done. The total diagonal Diag X m is dieomorphic to X and the simple diagonals in X n are the images of Diag(X m ) X n m under the factor permutation maps. The multi-diagonals (later called simply diagonals) are the intersections of these simple diagonals. Since we are assuming the oundary of X to e connected, we can identify a simple diagonal, involving equality for some collection of factors, with the oundary face B 2 B (2) which has a factor of @X exactly in each of the factors involving equality. Then the multi-diagonals D can e identied with transversally-intersecting susets B (2) ; meaning that dierent elements do not have a oundary factor in common. It is important to understand that the diagonals are not p-sumanifolds in X n : Nor, in general, are their oundary faces (which is what we are most interested in). Indeed there are always points analogous to the oundary of the diagonal in [0; 1] 2 and hence they do not have a local product structure consistent with that

SCATTERING CONFIGURATION SPACES 5 of the manifold. However, the eect of the construction aove is to resolve these `singularities'. (I.8) The lift from X n to X n of each D is a p-sumanifold Now, the lift of D will generally meet many oundary faces of X n : In particular it meets the lift of every oundary face B 2 B (2) with B \ and these intersections are the `oundary diagonals' which are to e lown up. Thus, (recalling that the lift of B 2 B (2) to X n is also denoted simply as B) set (I.9) H = fh B; = B \ D X n ; B 2 B with B \g: To low up all these sumanifolds we need to choose an order and this is required to respect the `lexicographic' partial ordering of H (I.10) H A;a 6 H B; () D a D or if D a = D then A B: Then the scattering conguration space X n sc is dened to e (I.11) X n sc = [X n ; H]: with low up in such an order. Of course it needs to e checked that the result is independent of the choice of order consistent with (I.10). As in the case of X n follows from the fact that the changes in order correspond to transversal intersections (including disjointness). As noted aove, these results are already known in the cases n = 2 and n = 3: Two new phenomena make the general case more complicated. The rst is the necessity to low-up multi-diagonals with the rst non-trivial multi-diagonal occuring for n = 4: The second is the issue of the ordering of H: I.3. Stretched projections. As noted aove, the existence of the `stretched projections' in (I.1) and (I.2) is the crucial property of the spaces Xsc: n The proof of their existence is in essence the same as that outlined aove, see (I.7), for the simpler n; maps ut the required commutation results are necessarily more intricate. In particular we need to consider various spaces intermediate etween X n and Xsc n which arise in this argument. For instance, the p-sumanifold H A;a X n actually makes sense already in [X n ; a] from which it can e lifted under further low up. These issues are discussed extensively in the article elow, here we ignore such niceties to explain the procedure used later. The notion of vertical and non-vertical for oundary faces with respect to the last factor discussed aove extends to the oundary diagonals H A;a : First extend it to transversal susets a B (2) where a is vertical if and only if \a is vertical. Thus H v;v = fh A;a 2 H; a and B are verticalg; (I.12) H nv;v = fh A;a 2 H; a is vertical ut B is non-verticalg; 2 H; a and B are non-verticalg: H nv;nv = fh A;a Notice that A \a in the denition of H A;a so if a is non-vertical, then so is A: From the denitions, it is clear that we have (I.13) X n 1 sc X = [X n ; B v ; H v;v ] with the appropriate order on the low ups. So the task is to recognize X n sc as an iterated low-up of this space. To do this, we rst show that all the `purely nonvertical' oundary diagonals can e lown down so that, as always with appropriate

6 RICHARD MELROSE AND MICHAEL SINGER orders on the collections of centres, X n sc = [X n ; H v;v [ H nv;v ; H nv;nv ] = [X n 1 X; B nv ; H v;v [ H nv;v ; H nv;nv ]: Thus all the oundary faces of non-vertical diagonals are rst removed. To proceed further, we remove the `last' (which means originally largest) oundary face from B 2 B nv y showing that it can e commuted past all the susequent oundary diagonals. Then all the H B;a corresponding to this oundary face are commuted out and lown down. Then this procedure is iterated, at each step removing the last remaining non-vertical oundary face and then the oundary diagonals contained in it. The rest of this article is devoted to providing a rigorous discussion of this outline. In x1 material on manifolds with corners and low up is riey recalled and in x2 the eect of the low up of oundary faces is considered. This is extended in x3 to get a asic result on the reordering of oundary low up, which is used extensively in the remainder of the article. In particular in x4 the results descried aove for the oundary conguration spaces X n are derived. The diagonals and their resolution is examined in x5 and the properties of these sumanifolds are slightly astracted in x6 to aid the discussion of iterative low up. Collections of oundary diagonals are descried in x7 and used to construct the spaces Xsc n in x8. Three results on the reordering of low ups of oundary diagonals are given in x9 and these are used to carry out the construction of the stretched projections in x10. A simple application of these spaces in x11 is inserted to indicate why these resolutions should prove useful. 1. Manifolds with corners Since we make heavy use of conventions for manifolds with corners, we give a rief description of the asic results which are used elow. These can also e found in the [3]. 1.1. Denition and oundary faces. By a manifold with corners we shall mean a space modelled locally on products [0; 1) k R n k with smooth transition maps (meaning they have smooth extensions across oundaries.) For such a space, M; C 1 (M) is well-dened y localization and at each point the oundary has a denite codimension, corresponding to the numer, k of functions in C 1 (M) vanishing at the point which are non-negative neary and have independent dierentials. We will insist that the oundary hypersurfaces, the closures of the sets of codimension 1; e emedded. This corresponds to the existence of functions i 2 C 1 (M) which are everywhere non-negative and have d i 6= 0 on f i = 0g and such that @M = f Q i i = 0g: The connected components of the sets f i = 0g are the oundary hypersurfaces, the collection of which is denoted M 1 (M): The components of the intersections of these hypersurfaces form oundary faces, all are closed and are the closures of their interiors the points of which have xed codimension; thus M k (M) consists of all the (connected) oundary faces of (interior) codimension k and M (k) (M) denotes the collection of codimension at least k: By convention, we shall include M 2 M(M) as a `oundary face of codimension zero'. Near a point of M; where the oundary has codimension k; it is generally natural to use coordinates adapted to the oundary. That is, local coordinates x i 0;

SCATTERING CONFIGURATION SPACES 7 i = 1; : : : ; k and y j ; j = 1; : : : ; n k where the oundary hypersurfaces through the given point are the fx i = 0g: If B 1 ; B 2 2 M(M) then their intersection is a oundary face (possily empty) ut their union is not. However the union is contained in a smallest oundary face which we will denote (1.1) B 1 u B 2 = \ fb 2 M(M); B B 1 [ B 2 g: 1.2. p-sumanifolds. Emedded sumanifolds of a manifold with corners can e rather more complicated locally than in the oundaryless case. The simplest type is a p-sumanifold. This is a closed suset Y M which has a local product decomposition near each point, consistent with a local product decomposition of M: An interior p-sumanifold (not necessarily contained in the interior) is distinguished y the fact that locally in a neighourhood U of each of its points there are l independent functions Z i 2 C 1 (U) which dene it and which are independent of the local oundary dening functions, i.e. it is dened y the vanishing of interior coordinates. A general p-sumanifold is an interior p-sumanifold of a oundary face. Any p-sumanifold Y of M can e locally put in standard form near a point p in the sense that there are adapted coordinates x i ; y j ; ased at p such that in the coordinate neighourhood U (1.2) U \ Y = f(x; y) 2 U; x j = 0; 1 j l; y i = c i for i 2 Ig where I is some suset of the index set for interior coordinates and the c i are constants. 1.3. Blow-up and lifting of manifolds and maps. The (radial) low up of a p- sumanifold is always well-dened and yields a new compact manifold with corners [M; Y ] with a low-down map : [M; Y ]! M which is a dieomorphism from the complement of the preimage of Y to the complement of Y: Lemma 1.1. Under low up of a oundary face of a manifold with corners, any p- sumanifold H lifts to a p-sumanifold which is contained in the lift of its oundaryhull, the smallest oundary face containing H: Note that the lift, also called the `proper transform', of a suset S M under the low up of a centre, B M; which is required to e p-sumanifold for this to make sense, is the suset of [M; B] which is either the inverse image 1 (S); if S B; or else the closure in [M; B] of 1 (S n B) if it is not. Of course this is a useful notion only for sets which are `well-placed' with respect to B: 1.4. Comparale and transversal sumanifolds. In the sequel the intersection properties of oundary faces, and later manifolds related to multi-diagonals, play an important role. The manifolds we consider here will always intersect cleanly, in the sense of Bott. That is, at each point of intersection they are modelled y linear spaces and their intersection is therefore also a manifold. This is immediate from the denition for oundary faces and almost equally ovious for the diagonal-like manifolds we consider later. We will say that two such manifolds, and this applies in particular to oundary faces, H 1 and H 2 ; of M are Comparale if H 1 H 2 or H 2 H 1 : Transversal, written H 1 t H 2 if N B 1 and N B 2 are linearly independent at each point of intersection. Neither comparale nor transversal, areviated to `n.c.n.t.' otherwise.

8 RICHARD MELROSE AND MICHAEL SINGER More generally a collection of sumanifolds H i ; i = 1; : : : ; J is transversal if at every point p of intersection of at least two of them, the conormals N p H i for those i for which p 2 H i are independent. This in particular implies that the intersection is a manifold. If B 1 t B 2 are two p-sumanifolds of M then the lift of B 2 to [M; B 1 ] which is dened aove to e the closure of B 2 n B 1 in the lown up manifold, is also equal to the inverse image, 1 (B 2 ): 1.5. -maps and -rations. A general class of maps etween manifolds with corners which leads to a category are the -maps. These are maps f : M! M 0 which are smooth in local coordinates and have the following additional property. Let 0 i e a complete collection of oundary dening functions on M 0 and j a similar collection on M: Then there should exist non-negative integers ij and positive functions a i 2 C 1 (M) such that (1.3) f 0 i = a i Y j ij j : Such a map is -normal if for each j; ij 6= 0 for at most one i: This means that no oundary hypersurfaces of M is mapped completely into a oundary face of codimension greater than 1 in M 0 : The real vector elds on M which exponentiate locally to dieomorphisms of M are the elements of V (M); meaning smooth vector elds on M which are tangent to all oundary faces. These form all the smooth sections of a natural vector undle T M over M and each -map has a -dierential at each point f : T p M! T f(p) M 0 : A -map is said to e a -sumersion if this map is always surjective. Blow-down maps are always -maps and for oundary faces they are -sumersions ut not for other p-sumanifolds. A -map which is oth a -sumersion and - normal is a -ration; low down maps are never -rations. With the notion of smoothness extended to include classical conormal functions on a manifold with corners, -rations are the analogues of rations in the sense that such regularity is preserved under push-forward. 2. Boundary low up Each oundary face B 2 M(M) of a manifold with corners is, as a consequence of the assumption that the oundary hypersurfaces are emedded, a p-sumanifold. Thus, it is always permissile to low it up. If B has codimension one (or zero), this does nothing. If k = codim(b) 2; i.e. B 2 M (2) (M); one gets a new manifold with corners, [M; B]; with one new oundary hypersurface ([M; B]) { which is the positive 2 k th part of a trivial (k 1)-sphere undle over B; k eing the codimension of B: This fractional-sphere undle is the inward-pointing part of the normal sphere undle to B and is trivialized y the choice of a dening function for each of the oundary hypersurfaces of M containing B: More generally, the other oundary hypersurfaces of [M; B] are in 1-1 correspondence with the oundary hypersurfaces of M where again the assumption that the oundary hypersurfaces are emedded means that the connectedness cannot change on low up of B: More generally, the oundary faces of [M; B] not contained in ([M; B]) are the lifts (closures of inverse images of complements w.r.t. B) of the oundary faces of M not contained in B: The oundary faces of [M; B] contained in ([M; B]) are either preimages (also

SCATTERING CONFIGURATION SPACES 9 called lifts) of oundary faces of B { hence are the restrictions of the fractionalsphere undles to oundary faces of B { or else are proper oundary faces of the fractional alls over one of these faces (including of course B itself). The latter ones are `new' oundary faces, not the lifts of old ones. We identify, at least notationally, each oundary face B 0 of M with its lift to a oundary face of [M; B]; even though the latter may e either a low-up of B 0 ; if B 0 is not contained in B initially, or a undle over B 0 if it is { in which case the dimension has increased. For later reference we examine the eect of the low up of a oundary face on the intersection of two others. Lemma 2.1. Consider two distinct oundary faces B 1 and B 2 in a manifold with corners M and their lifts to [M; B] where B 2 M (2) (M) : (2.1) (i) If B 1 t B 2 then their lifts are transversal in [M; B]; they are disjoint if and only if B 1 \ B 2 B ut B 1 n B 6= ; and B 2 n B 6= ;: (ii) If B 1 B 2 in M; then their lifts to [M; B] are never disjoint, they are comparale in [M; B] if and only if the lifts are transversal if B 1 n B 6= ;; B 1 B and B t B 2 or B 2 B; (2.2) B 1 B ( B 2 and are otherwise n.c.n.t. (that is if B = B 2 or B 1 B ut B and B 2 are neither transversal nor is B 2 B): (iii) If B 1 and B 2 are n.c.n.t. in M then their lifts are never comparale, are disjoint in [M; B] if and only if (2.3) B 1 \ B 2 B and oth B 1 n B 6= ; and B 2 n B 6= ;; they lift to meet transversally if and only if (using (1.1)) (2.4) either B 1 B ( B 1 u B 2 or B 2 B ( B 1 u B 2 and otherwise their lifts are n.c.n.t.. Proof. When discussing the local eect of the low up of a oundary face we may always choose adapted coordinates x i 0; y j ; in fact the `interior coordinates' y j play no part in the discussion here. There exist susets I, I 1, I 2 of f1; : : : ; kg such that (2.5) B i = fx j = 0 : j 2 I i g; i = 1; 2; B = fx j = 0 : j 2 Ig: The three parts of the Lemma correspond to the mutually exclusive cases where (i) I 1 \ I 2 = ;; (ii) I 1 I 2 or I 2 I 1 ; (iii) neither of these conditions hold. Of course we only need consider the rst of the cases in (ii). Near any point of the front face of [X; B] there are adapted coordinates with the interior coordinates, y j ; lifted from X, and the oundary coordinates x j replaced y X ( x j =t B if j 2 I (2.6) t B = x i ; and t j = x i2i j if j 62 I:

10 RICHARD MELROSE AND MICHAEL SINGER Note that (2.7) X j2i t j = 1: Here t B denes the new front face, i.e. the lift of B: The lifts ~ Bi of the B i to [X; B] are given y (2.8) ~ Bi = ( ftb = 0; t j = 0; j 2 I i n Ig if B i B () I I i ft j = 0; j 2 I i g if B i n B 6= ;; () I 6 I i : We can now examine the intersection properties of ~ B1 and ~ B2. (a) First assume B 1 and B 2 are oth contained in B. Then from (2.8) it is clear that the lifts ~ B1 and ~ B2 are never transversal, since t B ; the dening function of the front face, vanishes on the oth of them. They are clearly comparale if and only B 1 and B 2 are comparale and are otherwise n.c.n.t.. () Second, suppose B 1 B ut B 2 n B 6= ;. From (2.8) we see that ~ B1 and ~B 2 meet transversally if and only if I 1 n I and I 2 are disjoint. The lifts can only e comparale if ~ B1 ~ B2 and this occurs if and only if I 1 n I contains I 2 : Otherwise, ~ B1 and ~ B2 are n.c.n.t.. (c) Finally, suppose that B 1 n B and B 2 n B are oth non-empty. Then each ~B i is given y the vanishing of the t j for j 2 I i : Now I 1 [ I 2 I if and only if B 1 \ B 2 B: Hence ~ B1 and ~ B2 are disjoint in this case in view of (2.7). Otherwise, the transversal, comparale or n.c.n.t.according as this is the case for the original sumanifolds B 1 and B 2 : It is now a simple matter to use these oservations to prove the Lemma. Consider part (i) in which B 1 and B 2 are transversal, or equivalently, I 1 and I 2 are disjoint. Running through cases aove, (a) cannot arise and in () and (c) ~ B1 and ~ B2 are transversal and are disjoint exactly as claimed. Next consider part (ii) of the Lemma: without loss of generality, suppose B 1 B 2, so I 1 I 2 : Then in case (a) the lifts are comparale if oth are contained in B and according to (c) they are comparale if oth are not contained in B: Comparale lifts arise in case () if I 2 I 1 n I; or equivalently if I and I 2 are disjoint susets of I: Thus the lifts are comparale in this case if B 1 B 2 ; B 1 B; B and B 2 are transversal. We also see from () that the lifts of ~ B1 and ~ B2 are transversal if and only if I 1 \ I 2 n I \ I 2 = ; or equivalently if I 2 I: This proves (2.2). Finally consider part (iii) of the Lemma. Under case (a) the lifts are always n.c.n.t.. Under case (), the lifts are transversal if and only if (I 1 n I) \ I 2 = ;; that is, if and only if I 1 \I 2 I \I 2 : Since I I 1, this just means that I 1 \I 2 I which gives (2.4). Otherwise, they are n.c.n.t.. Under (c), the lifts are n.c.n.t.unless the B 1 \ B 2 B; in which case they are disjoint, giving (2.3). We are most interested in the transversality of the intersections of the lifts, meaning either they are disjoint or meet transversally. Corollary 2.2. If B 1 and B 2 are distinct oundary faces of a manifold with corners and B 2 M (2) (M) then B 1 and B 2 are (i.e. lift to e) transversal in [M; B] if and only if they are initially transversal or if not then (2.9) B 1 B ( B 1 u B 2 or B 2 B ( B 1 u B 2 :

SCATTERING CONFIGURATION SPACES 11 Two oundary faces lift to e disjoint if (2.10) B 1 \ B 2 B and oth B 1 n B 6= ; and B 2 n B 6= ;: Note that if B 1 t B 2 then B 1 u B 2 = M: 3. Intersection-orders Since oundary faces lift to oundary faces under low up of any oundary face, any collection, C M(M); in any manifold with corners, can e lown up in any preassigned order, leading to a well-dened manifold with corners. Of course this actually means that after the rst low-up the lift of the second oundary face is lown up, and so on. Let the order of low up e given y an injective function (3.1) o : C! N where for simplicity we also assume that the range is an interval [1; N] in the integers. Denote the total low-up as [M; C; o]; in general the choice of order does make a dierence to the nal result ut the interior is always canonically identied with the interior of M: From now on we will assume that the initial collection of oundary faces, C; of M which are to e lown up is closed under non-transversal intersection in M : (3.2) B; B 0 2 C M (2) (M) =) B t B 0 or B \ B 0 2 C: It should e noted that this is a condition in M and can fail for the lifts under low up of oundary faces in that the intersection of the lifts may not e equal to the lift of the intersection. Lemma 3.1. If C M(M) is closed under non-transversal intersection then it is a disjoint union of collections C i C which are also closed under nontransversal intersection, each contain a unique minimal element, and are such that all intersections etween elements of dierent C i are transversal. These C i may e called the transversal components of C: Proof. Consider the minimal elements A i 2 C; those which contain no other element. These must intersect transversally, since otherwise the intersection would e in C and they would not e minimal. Then set C i = ff 2 C; F A i g; these are certainly closed under non-transversal intersection. On the other hand the dening functions for elements of C i are amongst the dening functions for A i : It follows that the dierent C i are disjoint, since their elements cannot have a dening function in common, and also that intersections etween their elements are transversal. Denition 3.2. An order o on a collection C of oundary faces is an intersectionorder if for any pair B 1 and B 2 2 C which are not transversal or comparale, B 1 \ B 2 comes earlier than at least one of them, i.e. (3.3) B 1 ; B 2 2 C =) B 1 t B 2 or o(b 1 \ B 2 ) max(o(b 1 ); o(b) 2 )): Of course if B 1 and B 2 are comparale then the intersection is equal to one of them so (3.3) is automatic. On the other hand if B 1 and B 2 intersect nontransversally then (3.3) implies that the intersection comes strictly efore the second of them with respect to the order. We will not repeatedly say that C is closed under non-transversal intersection, just that it has an intersection-order which is taken to imply the closure condition.

12 RICHARD MELROSE AND MICHAEL SINGER Denition 3.3. An order o on a collection C of oundary faces is a size-order if the codimension is weakly decreasing with the order, i.e. (3.4) o(b 1 ) < o(b 2 ) =) codim(b 1 ) > codim(b 2 ): Clearly a size-order is an intersection-order since the intersection of n.c.n.t. oundary faces necessarily has larger codimension than either of them and so must occur rst in the order of the three. Lemma 3.4. The iterated low up in a manifold with corners M of a collection of oundary faces C; which is closed under non-transversal intersection, with respect to any two size orders gives canonically dieomorphic manifolds, with the dieomorphism eing the extension y continuity from the identications of the interiors. Proof. The rst element, B; in the order necessarily has maximal codimension so cannot contain any other. Thus all lifts of elements of C 0 = C n fbg are closures of complements with respect to B; their lifts therefore have the same dimension as efore and hence in the induced order on C 0 in [M; B] the codimension is weakly decreasing. Now, we proceed to show that the lift of the elements of C 0 to [M; B] is closed under non-transversal intersection. So, consider two distinct elements B 1 ; B 2 2 C 0 : If they are comparale then B cannot contain the smaller so y, Lemma 3.1 they lift to e comparale. If they are transversal then again y Lemma 2.1 they lift to e transversal. Finally, suppose B 1 and B 2 are n.c.n.t.. Since (2.4) cannot arise here, either (2.3) holds, and hence B 1 \ B 2 = B and they lift to e disjoint, or else B 1 \ B 2 n B 6= ; and they lift to e n.c.n.t. with intersection the lift of B 1 \ B 2 2 C 0 : Thus after the low up of the rst element of C the remaining elements lift to a collection of oundary faces closed under non-transversal intersection and in size-order. Now we can proceed y induction on the numer of elements of C and hence assume that we already know that the result of the low up of C 0 in [M; B] is independent of the size-order. If B is the only element of maximal codimension in C the result follows. If there are other elements of the same codimension then y Lemma 3.1 they meet B transversally. Thus, the order of B and the second element can e exchanged. Applying discussion aove twice it follows that the same manifold results from low up in any size-order on C: We proceed to show that the the same manifold results from the low up in any intersection-order. Proposition 3.5. The iterated low up of M; [M; C; o]; of an intersection-ordered collection of oundary faces is a manifold with corners independent of the choice of intersection-order in the sense that dierent orders give canonically dieomorphic manifolds, with the dieomorphism eing the extension y continuity from the identications of the interiors. Proof. Let o e the order in the form (3.1). For such an order we dene the defect to e (3.5) d(o) = X J2C o(j) maxf(codim(j) codim(i)) + ; o(i) < o(j)g: Here the codimensions are as oundary faces of M; not after low-up. Thus the defect is the sum over all sets of the maximum dierence (if positive) etween the codimensions of the `earlier' sets and of that set, weighted y the position of the

SCATTERING CONFIGURATION SPACES 13 set. Thus, for a size-order the defect vanishes, ecause all these dierences are non-positive, otherwise it is strictly positive. For a general intersection-order take the rst set, with respect to the order, I; such that its successor, J; had larger codimension in M; and consider the order o 0 otained y reversing the order of I and J: We claim that this is an intersectionorder and of strictly smaller defect, and that [M; C; o] = [M; C; o 0 ]. The last point will e checked rst. Certainly I cannot e the last element with respect to the order. Note also that the oundary faces up to, and including, I are in size-order, y the choice of I: Let P C e the sucollection of strict predecessors of I: Let M 0 e the manifold otained from M y lowing up P: In order to e ale to commute the lifts, I ~ and J; ~ of I and J to M 0 ; we need to rule out the possiility that they are n.c.n.t.; Lemma 2.1 will e used repeatedly for this. Suppose rst that I and J are comparale in M: Then J I: According to Lemma 2.1, such a comparale pair of sumanifolds can remain comparale, can ecome transversal, or can ecome n.c.n.t.in M 0 : If they ever ecome transversal, then they remain so under all susequent low-ups. Now comparale sumanifolds B 1 B 2 can only ecome n.c.n.t.under a low-up with centre B satisfying B 1 B = B 2 or B 1 B; B nb 2 6= ;. Since the centres of the low-ups leading to M 0 are all of smaller dimension than I; which here plays the role of B 2 ; we see that these conditions can never e met y elements B 2 P: So if I and J are comparale in M; their lifts to M 0 cannot e n.c.n.t.. The only remaining possiility is that I and J are n.c.n.t.in M: In this case, I \J has strictly dimension than I and so, ecause o is an intersection-order, this must e an element of P: Moreover, ecause P is in a size-order, the lifts of I and J meet in the lift of I \ J until this manifold is lown up, at which point they ecome disjoint and then remain disjoint under all susequent low-ups. This shows that ~I and J ~ commute in M 0 : To show that o 0 is an intersection-order, consider an n.c.n.t.pair A; B: The only possiility of a failure of the intersection-order condition for o 0 is if I was the intersection and J the second element, in the order, of such a pair. However this means that, initially in M; codim(i) > codim(j) and from the discussion aove, this cannot occur. Now, to compute the defect of o 0 oserve that each of the sets which came after J initially still have the same overall collection of sets preceding them, and the same order, hence make the same contriution to the defect. The same is true for the sets which preceded I: Thus we only need to recompute the contriutions from I and J after reversal. In its new position, J has one less precedent, viz. I now comes later, so the set of dierences codim(j) codim(i 0 ) where o(i 0 ) < o 0 (J) is smaller and the order of J has gone down, so it makes a strictly smaller contriution. The contriution of I was zero efore and is again zero, since the only extra set preceding it, namely J; has larger codimension than it. Thus this `move' strictly decreases the defect. Repeating the procedure a nite numer of times (note that after the rst rearrangement, J might well e the `new I') must reduce the defect to 0: Hence the low up for any intersection-order is (canonically) dieomorphic to one for which codim(b) is weakly decreasing, i.e. to a size-order and hence y Lemma 3.4 all intersection-orders lead to the same lown-up manifold.

14 RICHARD MELROSE AND MICHAEL SINGER Denition 3.6. We denote y [M; C] the iterated low-up of any collection of oundary faces which is closed under non-transversal intersection, with respect to any intersection-order. One simple rearrangement result which follows from this is: Lemma 3.7. Suppose C 1 C M(M) are oth closed under non-transversal intersection, then there is an intersection order on C in which the elements of C 1 come efore all elements of C n C 1 : Proof. Let o e a size-order on C and consider the new order o 0 on C dened y o 0 (B) = o(b) if 2 C 1 ; o 0 (B) = o(b) + N otherwise, where N = max(o): Then o 0 has the desired property that every element of C 1 comes efore every element of C n C 1 : Moreover, o 0 must e an intersection-order. Thus we wish to show that if B 1 and B 2 are n.c.n.t.then (3.6) o 0 (B 1 ) < o 0 (B 2 ) < o 0 (B 1 \ B 2 ) is not possile. This certainly cannot happen unless B 1 ; B 2 2 C 1 ; B 1 \ B 2 2 C n C 1 ; ecause o 0 restricts to give a size-order on each of C 1 and C n C 1 : However, if B 1 and B 2 lie in C 1 then so does B 1 \ B 2 ecause C 1 is closed under non-transversal intersection. Thus (3.6) is indeed impossile. Corollary 3.8. If C 1 C 2 are two collections of oundary faces of M; oth closed under non-transversal intersection, then there is an iterated low-down map (3.7) [M; C 2 ]! [M; C 1 ]: Proof. By the preceding lemma, there is an intersection-order on C 2 with respect to which all elements of C 1 come rst. The existence of the low-down map follows immediately from this. We will use the freedom to reorder low ups frequently elow. For instance if C is closed under non-transversal intersection then any given element is rst or last in some intersection-order. In fact if the elements are rst given a size-order then any one element can e moved to any other point in the order and the result is an intersection-order. Another use of the freedom to change order estalished aove is to examine the intersection properties of oundary faces, as in Lemma 2.1, ut after a sequence of oundary low ups. Proposition 3.9. If B 1 ; B 2 are distinct oundary faces of M and C M(M) is closed under non-transversal intersection then (1) The lifts of B 1 and B 2 to [M; C] are disjoint if they are disjoint in M or there exists B 2 C satisfying (2.10). (2) The lifts of B 1 and B 2 meet transversally in [M; C] if they are transversal in M or there exists B 2 B satisfying (2.9). (3) If B 1 B 2 in M then this remains true for the lifts to [M; C] if (3.8) B 2 C; B 1 B =) B 2 B or B 2 t B: Proof. We can assume that B 1 and B 2 are oth proper oundary faces. If there exists an element of C satisfying (2.10) then, as noted aove, there is an intersectionorder on C in which a given element comes rst. Lemma 2.1 shows that lowing it up rst separates B 1 and B 2 which thereafter must remain disjoint. Thus shows the suciency of 2.1.

SCATTERING CONFIGURATION SPACES 15 As aove, if there is an element of C satisfying (2.9) then it can e lown up rst in an intersection-order which makes B 1 and B 2 transversal; then Lemma 2.1 shows that persists under susequent low up. In the third part of the Proposition the suciency of the condition follows immediately from Lemma 2.1 since the elements of C of codimension two or greater containing B 1 ; and so y hypothesis either containing B 2 or transversal to it, form a collection closed under non-transversal intersection. In fact this is separately true of those containing B 2 and those which contain B 1 ut are transversal to B 2 since no intersection of the latter can contain B 2 : So all these low ups can e done rst. In each case once the minimal element of a transversal component is lown up the other elements do not contain B 1 so all low ups preserve the inclusion of B 1 in B 2 : Lemma 3.10. For any two oundary faces B 1 ; B 2 2 C; with lifts denoted Bi ~ ; i = 1; 2 it is always the case that (3.9) ~ B1 \ ~ B2 ^ B1 \ B 2 in [M; C]: Proof. Consider the decomposition C = C 0 [C 00 into the collections of elements which do not contain B 1 \ B 2 and those which do contain it. These must separately e closed under non-transversal intersection. Under low up of an element of C 0 ; B 1 ; B 2 and B 1 \ B 2 all lift to the closure of their complements with respect to the centre so the lift of the intersection is the intersection of the lifts. Moreover the other elements of C 0 lift not to contain the intersection while the elements of C 0 lift to contain it. Thus after lowing up all the elements of C 0 we are reduced to the case that C 0 is empty, so we may assume that B 1 \ B 2 is contained in each element of C: Now, consider the decomposition of C as in Lemma 3.1. Consider the eect of the low up of the minimal element A 1 2 C 1 : Now the lift of B 1 \ B 2 to [M; A 1 ] is its preimage, the lifts of B 1 and B 2 depend on whether they are, or are not, contained in A 1 ut in any case (3.9) holds after this single low up. If A 1 contains neither B 1 nor B 2 then y (2.10) the lifts are disjoint and we need go no further. On the other hand if A 1 B 1 [ B 2 then all three manifolds lift to their preimages and equality of intersection of lifts and the lift of the intersections persists. The lifts of the other elements of C 1 contain no res of the front face of [M; A 1 ] over A 1 and so cannot contain the intersection. Hence these low ups again preserve the equality. The only case remaining is where A 1 contains one, ut not oth, of B 1 and B 2 : We can assume that A 1 B 1 and then all the elements of C 1 satisfy this. Let C 1 = C1 0 [ C1 00 e the decomposition into those (efore low up of A 1 ) which do not contain B 2 and do contain B 2 ; where the second collection may e empty. Blowing up in size-order for each of these sucollections, oserve that after the low up of A 1 ; the other elements of C1 0 lift to the closures of their complements with respect to A 1 and hence cannot contain res of A 1 and hence cannot contain the lift of B 1 or the intersection. They intersection of the lifts in [M; A 1 ] is the intersection of the lift of B 2 and the front face. No other element of C1 0 can contain this, since then it would contain B 2 contrary to assumption. Thus the elements of C1 0 lifted to [M; A 1 ] do not contain the intersection of the lifts of B 1 and B 2 so after their low up the inclusion (3.9) still holds. On the other hand the elements of C1 00 do contain the lift of B 2 and continue to do so after all elements of C1 0 have een lown up. They therefore contain the intersection of the lifts ut cannot contain the lift of B 1 : Again C1 00 can e decomposed using Lemma 3.1 and

16 RICHARD MELROSE AND MICHAEL SINGER a minimal element can e lown up. For the elements which contain this minimal one the argument now proceed as for A 1 and C 1 aove, except that the lift of B 1 can never e contained in these centres. This means that (3.9) holds at the end of the low up of one of the transversal parts of C1 00 : However the other transversal components lift under these low ups to their preimages, so they contain the lift of B 2 ut not of B 1 and the argument can e repeated. Thus at the end of the low up of C 1 ; (3.9) holds. However the other transversal components of C again lift to their preimages so contain the intersection of the lift of B 1 and B 2 (and even the lift of the intersection). So the argument aove for C 1 can e repeated a nite numer of times to nally conclude that (3.9) remains true in [M; C]: 4. Boundary configuration spaces Let X e a compact manifold with oundary and consider M = X n for some n 2: The oundary faces of X n are just n-fold products with each factor either X or a component of its oundary. Dene B M (2) (X n ) to e equal to M (2) (X n ) if the oundary of X is connected, otherwise to e the proper suset consisting of those n-fold products where each factor is either X or the same component of the oundary in the remaining factors, and where there are at least two of these factors. Lemma 4.1. The collection B M(X n ) is closed under non-transversal intersection. Proof. The intersection of two elements where all oundary factors arise from the same oundary component of X are certainly in B : So consider two elements B 1 ; B 2 of B with dierent oundary components, A 1 X for the rst and A 2 X for the second. Then A 1 \ A 2 = ;; since X is a manifold with oundary, so has no corners. Thus if dierent oundary components occur in any one factor in B 1 and B 2 then B 1 \ B 2 = ;: The only remaining case is when each oundary factor in B 1 corresponds to a factor of X in B 2 and then the intersection is transversal. Thus B is closed under non-transversal intersection. Denition 4.2. The n-fold -stretched product of X is dened to e (4.1) X n = [X n ; B ]: This denition relies on Proposition 3.5 and Denition 3.6 to make it meaningful. Boundary faces of codimension one, or indeed the whole of X n ; could e included since low up of these `oundary faces' is to e interpreted as the trivial operation. Remark 4.3. We will generally concentrate on the case that X has one oundary component so (4.1) amounts to lowing up all the oundary faces; in this case B = B (2) : Even if the oundary of X is not connected then lowing up all elements of B (2) = M (2) (X n ); in an intersection order, is perfectly possile. The result may e called the `overlown' product (4.2) X n o = [X n ; B (2) (X n )]; @X not connected. Since we are mainly interested in considering the resolution of diagonals, the smaller manifold in (4.1) is more appropriate here. Next we give a more signicant application of Proposition 3.5.

SCATTERING CONFIGURATION SPACES 17 Proposition 4.4. If m < n; each of the projections o n m factors of X; : X n! X m ; xes a unique `-stretched projection' giving a commutative diagramme (4.3) X n X m and furthermore is a -ration. X n X m Proof. The existence of follows from Corollary 3.8. Namely, taking to e the projection o the last n m factors for simplicity of notation, the sucollection of B v() for X n ; consisting of the oundary faces of X n in which the last n m factors consist of X; is closed under non-transversal intersection. Thus, using Corollary 3.8, there is an iterated low-down map (4.4) f : [X n ; B ]! [X m ; B X n m ]: Composing this with projection o the last n m factors gives a map for which the diagramme (4.3) commutes. Since oth the iterated low-down map and the projection are -sumersions, so is : To see that it is a -ration it suces to show that each oundary hypersurface of X n is mapped into either a oundary hypersurface of X m or onto the whole manifold; this is `-normality'. As a -map maps each oundary face into a oundary face so it is enough to see what happens near the interior of each oundary hypersurface of X n : If the oundary hypersurface in question is not the result of some low up then looks locally the same as and local -normality follows. If it is the result of low up then maps into the interior provided the oundary face is not the lift of a oundary face, necessarily of codimension two or greater, from X m : If it is such a lift then is locally the projection onto X m ; i.e. maps into the interior of the corresponding front face. We shall analyze more fully the structure of the oundary faces of [X n ; C] where C B is some collection closed under non-transversal intersection. Unless otherwise stated elow, although mostly for notational reasons, we will make the simplifying restriction that (4.5) The oundary of X is connected so B = B (2) = M (2) (X n ): For a oundary face B 2 B (2) it is convenient to consider three distinct possiilities (i) B 2 C (ii) B =2 C ut there exists A 2 C; B A: (iii) B =2 C and A B =) A =2 C: In the rst case (4.6) C = fbg [ Sm(B) [ Bi(B) [ Nc(B) is a disjoint union, where (4.7) Sm(B) = fb 0 2 C; B 0 $ Bg Bi(B) = fb 0 2 C; B 0 % Bg Nc(B) = fb 0 2 C; B and B 0 are not comparaleg: