CALCULUS OF FUNCTORS, OPERAD FORMALITY, AND RATIONAL HOMOLOGY OF EMBEDDING SPACES GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C ____ Abstract.Let M be a smooth manifold and V a Euclidean space. Let Emb(M, * *V ) be the homotopy_fiber_of the map Emb(M, V ) -! Imm(M, V ). This paper is about t* *he rational homol- ogy of Emb(M, V ). We study_it_by applying embedding calculus and orthogo* *nal calculus to the bi-functor (M, V ) 7! HQ ^ Emb(M, V )+. Our main theorem states that if d* *imV 2 ED(M) + 1 (where ED(M) is the embedding dimension of M), the Taylor tower in the se* *nse of orthogonal calculus (henceforward called "the orthogonal tower") of this functor spl* *its as a product of its layers. Equivalently, the rational homology spectral sequence associated * *with the tower collapses at E1. In the case of knot embeddings, this spectral sequence coincides w* *ith the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Ko* *ntsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of t* *he functor ____ HQ ^ Emb(M, V )+. The formulas show, in particular, that the (rational) homotopy type of th* *e layers of the or- thogonal tower is determined by the (rational) homotopy type of M. This, * *together with our rational splitting theorem,_implies that under the above assumption on co* *dimension, the rational homology groups of Emb(M, V ) are determined by the rational homotopy typ* *e of M. Contents 1. Introduction * * 2 1.1. A section by section outline * * 6 1.2. Acknowledgments * * 6 2. Spaces, spectra, and chain complexes * * 6 2.1. Postnikov sections * * 7 2.2. Diagrams * * 8 2.3. Homotopy limits * * 8 3. Formality and homogeneous splitting of diagrams * * 8 4. Enriched categories and their modules * * 9 4.1. Monoidal model categories and enriched categories * * 9 4.2. Enriched categories * * 10 4.3. Homotopy theory of right modules over enriched categories * * 11 4.4. Lax monoidal functors, enriched categories, and their modules * * 12 ___________ 1991 Mathematics Subject Classification. Primary: 57N35; Secondary: 55P62, 55* *T99. Key words and phrases. calculus of functors, embedding calculus, orthogonal c* *alculus, embedding spaces, operad formality. The first and third authors were supported by the National Science Foundation* *, grants DMS 0605073 and DMS 0504390 respectively. 1 2 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C 4.5. Discretization of enriched categories * * 13 5. Operads and associated enriched categories * * 14 5.1. Operads and right modules * * 14 5.2. Enriched category associated to an operad * * 14 5.3. The standard little balls operad * * 15 6. Formality and splitting of the little balls operad * * 16 7. Formality of a certain diagram arising from embedding calculus * * 18 8. More generalities on calculus of functors * * 20 8.1. Embedding calculus * * 20 8.2. Orthogonal calculus * * 22 8.3. Bifunctors * * 22 9. Formality and the embedding tower * * 23 10. Formality and the splitting of the orthogonal tower * * 25 11. The layers of the orthogonal tower * * 29 References * * 32 1.Introduction Let M be a smooth manifold of dimension m. M may be non-compact, but we alway* *s assume that M is the interior of a compact manifold with boundary. Let V be a Euclidea* *n space. Let Emb (M, V ) be the space of smooth embeddings of M into V . For technical reaso* *ns, rather than study Emb (M, V ) directly, we will focus on the space ____ Emb (M, V ) := hofiber(Emb(M, V ) -! Imm (M, V,)) where Imm (M, V ) denotes the space of immersions of M into V . Note that the d* *efinition requires that we fix an embedding (or at minimum an immersion) ff : M ,!_V_, to act as a* * basepoint. Most of the time we will work with the suspension spectrum__1_Emb (M, V )+, and* *_our_results are really about the rationalization of this spectrum, 1QEmb_(M,_V )+ ' HQ ^ E* *mb (M, V )+. In other words, our results are about the rational homology of Emb (M, V ). Our framework is provided by the Goodwillie-Weiss calculus of functors. One * *of the main features of calculus of functors is that it associates to a functor a tower of * *fibrations, analogous to the Taylor series of a function. The functor Emb (M, V ) is a functor of tw* *o variables, and accordingly one may do "Taylor expansion" in at least two ways: In either the v* *ariable M or the variable V (or both). Since the two variables of Emb (M, V ) are of rather diff* *erent nature (for example, one is contravariant and the other one is covariant), there are two ve* *rsions of calculus needed for dealing with them - embedding calculus (for the variable M) and orth* *ogonal calculus (for the variable V ). Embedding calculus [24, 11] is designed for studying contravariant isotopy fu* *nctors (co-functors) on manifolds, such as F (M) = Emb (M, V ). To a suitable cofunctor F , embeddin* *g calculus asso- ciates a tower of fibrations under F (1) F (-) -! T1 F (-) -! . .-.! TkF (-) -! Tk-1F (-) -! . .-.! T1F (-) . CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 3 Here TkF (U) := holim F (U0), {U02Ok(M)|U0 U} where Ok(M) is the category of open subsets of M that are homeomorphic to the d* *isjoint union of at most k open balls. T1 F is defined to be the homotopy inverse limit of TkF . When circumstances * *are favorable, the natural map F (M) ! T1 F (M) is a homotopy equivalence, and then one says that * *the embedding tower converges. There is a deep and important convergence result, due to Goodw* *illie and Klein (unpublished, see [9]), for the functor F (M) = Emb (M, N), where N is a fixed * *manifold. We will state it now, it being an important fact in the background, but we will not rea* *lly use it in this paper. Theorem 1.1 (Goodwillie-Klein,_[9]). The Taylor tower (as defined above) of the* * embedding functor Emb (M, N) (or Emb (M, N)) converges if dim(N) - dim(M) 3. We will only need a much weaker convergence result, whose proof is accordingl* *y easier. The "weak convergence theorem" says that the above Taylor tower converges if 2 dim(* *M)+2 < dim(N) and a proof can be found_in the remark after Corollary 4.2.4 in [10]. The weak * *convergence result also holds for HQ ^ Emb (M, N)+ by the main result of [25]._ Let us have a closer look at the cofunctor U 7! HQ ^ Emb (U, V )+. If_U_is_h* *omeomorphic to a disjoint union of finitely many open balls, say U ~=kU x Dm , then Emb (U,* * V ) is homotopy equivalent to the configuration space C(kU, V ) of kU-tuples of distinct points* * in V or, equivalently, the space of kU-tuples of disjoint balls in V , which we denote B(kU, V ). Abus* *ing notation slightly, we can write that ____ ____ (2) TkHQ ^ Emb (M, V )+ := holim HQ ^ Emb (U, V )+ ' holim HQ ^ B(kU, V )+ U2Ok(M) U2Ok(M) The right hand side in the above formula is not really well-defined, because B(* *kU, V ) is not a functor on Ok(M), but it gives the right idea. The formula tells_us_that und* *er favorable circumstances (e.g., if 2 dim(M) + 2 < dim(V )), the spectrum HQ ^ Emb(M, V )+ * *can be written as a homotopy inverse limit of spectra of the form HQ ^ B(kU, V )+. It is obvio* *us that the maps in the diagram are closely related to the structure map in the little balls ope* *rad. Therefore, information about the rational homotopy type of the little balls operad may yie* *ld information about the homotopy type of spaces of embeddings. The key fact about the little* * balls operad that we want to use is the theorem of Kontsevich ([14, Theorem 2 in Section 3.2* *]), asserting that this operad is formal. Theorem 1.2 (Kontsevich, [14]). The little balls operad {B (n, V )}n 0 is forma* *l over the reals. In other words, there is a chain of quasi-isomorphisms of operads of chain comp* *lexes connecting the operads C*(B (n, V )) R and H*(B (n, V ); R). The formality theorem was announced by Kontsevich in [14], and an outline of * *a proof was given there. However, not all the steps of the proof are given in [14] in as much det* *ail as some readers might perhaps wish. Because of this, the second and the third author decided to* * write another paper [16], whose primary purpose is to provide a complete and detailed proof o* *f the formality theorem, following Kontsevich's outline. The paper [16] also has a second purp* *ose, which is to prove a slight strengthening of the formality theorem, which we call "a rela* *tive version" of the formality theorem (Theorem 6.1 in the paper). We will give a sketch of the* * proof of the 4 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C relative version in Section 6. Using the relative version of formality, togethe* *r with some abstract homotopy theory, we deduce our first theorem (see Theorem 7.2 for a precise sta* *tement). Theorem 1.3. Suppose that the basepoint embedding ff : M ,! V factors through * *a vector subspace W V such that dim(V ) 2 dim(W ) + 1. Then the functor ____ U 7! C*(Emb (U, V )) R is a formal diagram of real chain complexes. This means that there is a chain o* *f weak equivalences, natural in U ____ ____ C *(Emb (U, V )) R ' H*(Emb (U, V ); R) To be precise, in the above theorem the domain over which U ranges is a certa* *in category Ofsk(M), which is closely related to Ok(M) (where k can be arbitrarily large). * *For the duration of the introduction, we will pretend that the two categories are the same. The* * basic idea in proving the theorem is to think of operads as enriched categories, and to inter* *pret the formality of the little balls operad as the formality of a certain_enriched_functor. Then* * we show that the functor from Ok(M) to chain complexes given by U 7! C*(Emb (U, V )+) factors, u* *p to a suitable notion of equivalence, through this formal functor, and therefore it, too, must* * be formal. To make all this work, we will have to invoke a fair amount of abstract homotopy t* *heory (Quillen module structures, enriched categories, etc). In particular, we will use some r* *esults of Schwede and Shipley [20] on the homotopy theory of enriched categories. A formality theorem similar to Theorem 1.3 was used in [17] for showing the c* *ollapse (at E2) of a certain spectral sequence associated to the embedding tower for spaces of * *knot embeddings. However, to obtain a collapsing result for a spectral sequence for more general* * embedding spaces, we need, curiously enough, to turn to Weiss' orthogonal calculus (the standard * *reference is [23], and a brief overview can be found in Section 8). This is a calculus of covaria* *nt functors from the category of vector spaces and linear isometric inclusions to topological sp* *aces (or spectra). To such a functor G, orthogonal calculus associates a tower of fibrations of fu* *nctors PnG(V ), where PnG is the n-th Taylor polynomial of G in the orthogonal sense. Let DnG(V* * ) denote the n-th homogeneous layer in the orthogonal Taylor tower, namely the fiber of the * *map PnG(V ) ! Pn-1G(V ). ____ The functor that we care about_is,_of course, G(V ) = HQ_^_Emb (M, V )+ where* * M is fixed. We will use the notation PnHQ ^ Emb (M, V )+ and DnHQ ^ Emb (M, V )+ to denote * *its Taylor approximations and homogeneous layers in the sense of orthogonal calculus. It * *turns out that Theorem 1.3_implies that, under the same condition on the codimension, the orth* *ogonal tower of HQ ^ Emb (M, V )+ splits as a product of its layers. The following is our m* *ain theorem (Theorem 10.6 in the paper). Theorem 1.4. Under the assumptions of Theorem 1.3, there is a homotopy equivale* *nce, natural with respect to embeddings in the M-variable (note that we do not claim that th* *e splitting is natural in V ) ____ Yn ____ PnHQ ^ Emb (M, V )+ ' DiHQ ^ Emb (M, V )+. i=0 The following corollary is just a reformulation of the theorem. CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 5 Corollary_1.5._Under the assumptions of Theorem 1.3 and Theorem 1.4, the spectr* *al sequence for H*(Emb_(M, V ); Q) that arises from the Taylor tower (in the sense of ortho* *gonal calculus) of HQ ^ Emb (M, V )+ collapses at E1. Here is a_sketch_of the proof of Theorem 1.4. Embedding calculus tells us, ro* *ughly speaking, that HQ ^ Emb (M, V )+ can be written as a homotopy limit of a diagram of spect* *ra of the from HQ ^ B(k, V )+. Since there is a Quillen equivalence between the categories of * *rational spectra and rational chain complexes, we may pass to a diagram_of rational chain comple* *xes of the form C *(B (k, V )) Q, whose homotopy limit is C*(Emb (M, V )) Q. On the other h* *and, Theorem 1.3 tells us that this diagram of chain complexes is formal when tensored with R. I* *t turns out that in our_case_tensoring with R commutes with taking the homotopy limit, and one c* *oncludes that C *(Emb (M, V )) R splits as the product of inverse limits of layers in the_P* *ostnikov_towers of C *(B (k, V )) R._It_follows that there must be a similar splitting for C*(Em* *b (M, V )) Q and therefore for HQ ^ Emb (M, V )+. On the other hand, it turns out that for funct* *ors of the form HQ ^ B(k, V )+,_the_Postnikov tower coincides, up to regrading, with the orthog* *onal tower, and therefore HQ^Emb (M, V )+ splits as the product of inverse limits of layers in * *the orthogonal tower of rationally stabilized configuration spaces. But, taking the n-th layer in th* *e orthogonal tower is an operation that commutes (in our case) with homotopy inverse limits_(unlik* *e the operation of taking the n-th layer of the Postnikov tower), and therefore HQ ^ Emb (M, V * *)+ splits as the product of layers of its orthogonal tower. Remark 1.6. In the case of knot embeddings, the spectral sequence associated wi* *th the orthogonal tower coincides with the famous spectral sequence constructed by Vassiliev, sin* *ce the latter also collapses, and the initial terms are isomorphic. This will be discussed in more* * detail in [17]. ____ In Section 11 we write an explicit description of Dn 1 Emb (M, V )+, in terms* * of certain spaces of partitions (which can also be described as spaces of rooted trees) attached * *to M. One purpose of Section 11 is to provide a motivation and a wider context for the rest of th* *e paper. This section is an announcement; detailed proofs will appear in [1]. We do note the followi* *ng consequence of our description of the layers: The homotopy groups of the layers depend only* * on the stable homotopy type of M and similarly the rational homotopy groups of the layers dep* *end only on the rational stable homotopy type of M (Corollary 11.2). Combining this with Th* *eorem 1.4, we obtain the following theorem (Theorem 11.6 in the paper). Theorem_1.7._Under the assumptions of Theorem 1.4, the rational homology groups* * of the space Emb (M, V ) are determined by the rational homology type of M. More preci* *sely, suppose M1, M2, V satisfy the assumptions of Theorem 1.4, and suppose there is a zig-za* *g of maps, each inducing an isomorphism in rational homology, connecting M1 and M2. Then there* * is an iso- morphism ____ ____ H *(Emb (M1, V ); Q) ~=H*(Emb (M2, V ); Q). In view of this result,_one may wonder whether the rational homotopy type (ra* *ther than just rational homology) of Emb (M, V ) could be an invariant of the rational homotop* *y type of M (in high enough codimension). One could derive further hope from the fact that * *the little balls operad is not only formal,_but also coformal. We will approach this question f* *or the rational homotopy groups of Emb (M, V ), at least in the case of knots, in [3]. A general point that we are trying to make with this paper is this: while emb* *edding calculus is important, and is in some ways easier to understand than orthogonal calculus, t* *he Taylor tower 6 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C in the sense of orthogonal calculus is also interesting and is worthy of a furt* *her study. We hope that Section 11 will convince the reader that the layers of the orthogonal towe* *r, while not exactly simple, are interesting, and it may be possible to do calculations with them. W* *e hope to come back to this in the future. 1.1. A section by section outline. In Section 2 we review background material a* *nd fix ter- minology on spaces, spectra and chain complexes. In Section 3 we define the not* *ion of formality of diagrams chain complexes. The main result of this section is the following s* *imple but useful observation: the stable formality of a diagram can be interpreted as the splitt* *ing of its Postnikov tower. Our next goal is to exploit Kontsevich's formality of the little balls operad* *s and deduce some formality results of diagrams of embedding spaces. In order to do that we first* * review, in Section 4, enriched categories, their modules and the associated homotopy theory. In Secti* *on 5 we review classical operads and their modules and give an alternative viewpoint on those * *in terms of enriched categories. This will be useful for the study of the homotopy theory of modules* * over an operad. We then digress in Section 6 to prove a relative version of Kontsevich's formal* *ity of the little balls operads that we need for our applications. In Section 7 we deduce the for* *mality of a certain diagram of real-valued chains on embedding spaces. In Section 8 we digress again to give a review of embedding calculus and orth* *ogonal calculus, and record some generalities on how these two brands of calculus may interact. * * In Section 9 we use the formality of a diagram of chains on embedding_spaces established in * *Section 7 to show that the stages in the embedding tower of HQ ^ Emb (M, V )+ split in a cer* *tain way, but not as the product of the layers in the embedding tower. In Section 10 we rein* *terpret this splitting once again, to_prove_our main theorem: Under a certain co-dimension h* *ypotheses, the orthogonal tower of HQ^Emb (M, V )+ splits as the product of its layers. In Sec* *tion 11 we sketch a description of the layers in the orthogonal tower, and deduce that the ration* *al homology of the space of embeddings (modulo immersions) of a manifold into a high-dimensional v* *ector space is determined by the rational homology type of the manifold. 1.2. Acknowledgments. The second author thanks Enrico Vitale for help with enri* *ched cate- gories. 2.Spaces, spectra, and chain complexes Let us introduce the basic categories that we will work with. o Top will stand for the category of compactly generated spaces (we choose co* *mpactly generated to make it a closed monoidal category, see Section 4). If X is a space we denot* *e by X+ the based space obtained by adjoining a disjoint basepoint. o Spectrawill be the category of (-1)-connected spectra. We denote by HQ the* * Eilenberg- MacLane spectrum such that ss0(HQ) = Q. A rational spectrum is a module spectru* *m over HQ. For a space X, 1 X+ stands for the suspension spectrum of X, and HQ^X+ denotes* * the stable rationalization of X. It is well-known that there is a rational equivalence HQ * *^ X+ ' o1pX+. o V will denote the category of rational vector spaces (or Q-vector spaces),* * and V the category of simplicial Q-vector spaces. o ChQ and Ch Rwill denote the category of non-negatively graded rational and* * real chain complexes respectively. We will some times use Ch to denote either one of these* * two categories. CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 7 Most of the above categories have a Quillen model structure, which means that o* *ne can apply to them the techniques of homotopy theory. A good introduction to closed model* * categories is [8], a good reference is [13]. There are slight variations in the literatur* *e as to the precise definition of model structure. We use the definition given in [13]. In particul* *ar, we assume the existence of functorial fibrant and cofibrant replacements. The category for w* *hich we will use the model structure most heavily is the category of chain complexes. Thus we re* *mind the reader that the category of chain complexes over a field has a model structure where w* *eak equivalences are quasi-isomorphisms, fibrations are chain maps that are surjective in positi* *ve degrees, and cofibrations are (since all modules are projective) chain maps that are injecti* *ve in all degrees [8, Theorem 7.2]. We will also need the fact that the category of rational spectra * *is a Quillen model category and is Quillen equivalent to the category ChQ . For a proof of this (i* *n fact, of a more general statement, involving the category of module spectra over a general Eile* *nberg - Mac Lane commutative ring-spectrum) see, for example, [19]. We now define some basic functors between the various categories in which we * *want to do homotopy theory. 2.0.1. Homology. We think of homology as a functor from chain complexes to chai* *n complexes. Thus if C is a chain complex, then H *(C) is the chain complex whose chain grou* *ps are the homology groups of C, and whose differentials are zero. Moreover, we define Hn(* *C) to be the chain complex having the n-th homology group of C in degree n and zero in all o* *ther degrees. Thus, H nis a functor from Ch to Ch as well. Notice that there are obvious iso* *morphisms of functors 1 M 1Y H*~= Hn ~= Hn. n=0 n=0 2.0.2. The normalized chains functor. To get from spaces to chain complexes, we* * will use the normalized singular chains functor C* : Top ! Ch, defined as C*(X) = N(Q[So(X)]). Here So(X) is the simplicial set of singularosimplicespof X, Q[So(X)] is the si* *mplicial Q-vector space generated by So(X), and N : V ! Ch is the normalized chains functor as* * defined for example in [22, Chapter 8]. 2.1. Postnikov sections. We will need to use Postnikov towers in the categories* * of chain com- plexes, and spectra. We now review the construction of Postnikov towers in the* * category of chain complexes. For an integer n and a chain complex (C, d), let d(Cn+1) be th* *e n-dimensional boundaries in C. We define the nth-Postnikov section of C, denoted (Pon(C), d0)* *, as follows 8 >: 0 if i > n +,1 The differential d0is defined to be d in degrees n, and the obvious inclusion* * d(Cn+1) ,! Cn in degree n+1. It is easy to see that Pon defines a functor from Ch to Ch. Moreove* *r, Hi(Pon(C)) ~= H i(C) for i n and Hi(Pon(C)) = 0 for i > n. For each n, there is a natural fibration (i.e., a degree-wise surjection) ssn* *: Pon(C) i Pon-1(C) defined as follows: ssn is the identity in all degrees except n + 1 and n; in d* *egree n + 1 it is the 8 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C zero homomorphism; and in degree n it is the obvious surjective map d : Cn ! d(* *Cn). Since ssn is a fibration, ker(ssn) can serve as the model for its homotopy fiber. Clearly* *, ker(ssn) is a chain complex concentrated in dimensions n and n + 1. The homology of the kernel is c* *oncentrated in dimension n, and in this dimension it equals the homology of the original compl* *ex C. A similar formula defines a natural map aen: C ! Pon(C), and we have ssnaen = aen-1. Note* * that aen, like ssn+1, is an isomorphism (on chain level) in degrees n. 2.2. Diagrams. Let A be a small category and let E be a category. An A-diagram * *in E is just a functor F :A ! E. In this paper a diagram can be a functor which is either c* *ovariant or contravariant. A morphism of A-diagrams is a natural transformation between tw* *o functors. Such a morphism is called a weak equivalence if it is a weak equivalence object* *wise, for a given notion of weak equivalence in the category E. In practice, we will only consid* *er diagrams of spaces, chain complexes or spectra. 2.3. Homotopy limits. We will make heavy use of homotopy limits of diagrams in * *Spectraand in Ch. Homotopy limits of diagrams in a general model category are treated in [* *13], Chapter 19. Generally, when we take the homotopy limit of a diagram, we assume that all the* * objects in the diagram are fibrant and cofibrant - this will ensure "correct" homotopical beha* *vior in all cases. Since most of our homotopy limits will be taken the category of chain complexes* * over Q or R, in which all objects are fibrant and cofibrant, this is a moot point in many cases* *. The only other category in which we will take homotopy limits is the category of rational spec* *tra, in which case we generally assume that we have taken fibrant-cofibrant replacement of all obj* *ects, whenever necessary. It follows from the results in [13], Section 19.4, that if R and L are the ri* *ght and left adjoint in a Quillen equivalence, then both R and L commute with homotopy limits up to * *a zig-zag of natural weak equivalences. In particular, this enables us to shuttle back a* *nd forth between homotopy limits of diagrams of rational spectra and diagrams of rational chain * *complexes. 3. Formality and homogeneous splitting of diagrams The notion of formality was first introduced by Sullivan in the context of ra* *tional homotopy theory [21, 7]. Roughly speaking a chain complex (possibly with additional stru* *cture) is called formal if it is weakly equivalent to its homology. In this paper we will only * *use the notion of formality of diagrams of chain complexes (over Q and over R). Definition 3.1. Let A be a small category. An A-diagram of chain complexes, F :* *A ! Ch, is formal if there is a chain of weak equivalences F ' H*OF . Formality of chain complexes has a convenient interpretation as the splitting* * of the Postnikov tower. Definition 3.2. Let A be a small category. We say that an A-diagram of chain c* *omplexes, F :A ! Ch, splits homogeneously if there exist A-diagrams {Fn}n2N of chain comp* *lexes such that F ' nFn and H*(Fn) = Hn(Fn) (i.e., Fn is homologically concentrated in de* *gree n). Proposition 3.3. Let A be a small category. An A-diagram of chain complexes is * *formal if and only if it splits homogeneously. CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 9 Proof.Let F be an A-diagram of chain complexes. In one direction, if F is formal then F ' H*(F ). Since H* = n2N Hn, we get * *the homogeneous splitting F ' n Hn(F ). In the other direction, suppose that F ' n2NFn with H*(Fn) = Hn(Fn) = Hn(F )* *. Recall the definition of Postnikov sections of chain complexes from Section 2. Then i ssn j ker Po n(Fn) i Pon-1(Fn) is concentrated in degrees n and n + 1 and its homology is exactly Hn(F ). Thus* * we have a chain of quasi-isomorphisms Fn -'!Po n(Fn) -' ker(Pon(Fn) ! Pon-1(Fn))-'!H n(ker(Pon(Fn) ! Pon-1(Fn)))~=Hn* *(F ), and so F ' n Hn(F ) = H*(F ). Remark 3.4. Note that in the above we proved the following (elementary) stateme* *nt: Suppose F and G are two A-diagrams of chain complexes such that both F and G are homolo* *gically concentrated in degree n and such that there is an isomorphism of diagrams Hn(F* * ) ~=H n(G). Then there is a chain of weak equivalences, F ' G. Using the Quillen equivalenc* *e between rational spectra and rational chain complexes, one can prove the analogous statement for* * diagrams of Eilenberg-Mac Lane spectra: If F and G are two A-diagrams of Eilenberg-Mac Lan* *e spectra concentrated in degree n, and if there is an isomorphism of diagrams ssn(F ) ~=* *ssn(G) then there is a chain of weak equivalences F ' G. Remark 3.5. Let F be a diagram with values in Ch. There is a tower of fibration* *s converging to holimF whose n-th stage is holimPonF . We call it the lim-Postnikov tower. O* *fQcourse, this tower does not usually coincide with the Postnikov tower of holimF . Since H* ~* *= 1n=0Hn, and homotopy limits commute with products, it follows immediately that if F is a fo* *rmal diagram then the lim-Postnikov tower of holimF splits as a product, namely 1Y holimF ' holimHn OF n=0 The proof of the following is also straightforward. Lemma 3.6. Let ~: A ! A0be a functor between small categories and let F be an A* *0-diagram of chain complexes. If the A0-diagram F is formal then so is the A-diagram ~*(F* * ) := F O ~. 4.Enriched categories and their modules We now briefly recall some definitions and facts about symmetric monoidal cat* *egories, enriched categories, Quillen module structures, etc. The standard reference for symmetri* *c monoidal cate- gories and enriched categories is [5, Chapter 6]. We will also need some result* *s of Schwede and Shipley on the homotopy theory of enriched categories developed in [20], especi* *ally Section 6, which is where we also borrow some of our notation and terminology from. 4.1. Monoidal model categories and enriched categories. A closed symmetric mono* *idal category is a triple (C, , 1) such that and 1 endows the category C with a s* *ymmetric monoidal structure, and such that, for each object Y , the endofunctor - Y :C ! C , X 7!* * X Y admits a right adjoint denoted by C(Y, -): Z 7! C(Y, Z). It is customary to think of C(Y* *, Z) as an "internal mapping object". Throughout this section, C stands for a closed symmetric monoi* *dal category. 10 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C A monoidal model category is a closed symmetric monoidal category equipped wi* *th a compatible Quillen model structure (see [20, Definition 3.1] for a precise definition). The only examples of monoidal model categories that we will consider in this * *paper are (1) The category (Top, x, *) of compactly generated topological spaces with * *cartesian prod- uct; (2) The category (Ch , , K) of non-negatively graded chain complexes over K* * (where K is Q or R), with tensor product. The internal hom functor in the category Ch is defined as follows. Let Y*, Z* b* *e chain complexes. Then Ch(Y*, Z*) is the chain complex that in positive degrees p > 0 is defined * *by 1Y Chp(Y*, Z*) = hom(Yn, Zn+p) n=0 while in degree zero, we have Ch0(Y*, Z*) = {chain homomorphisms fromY* toZ*}. The differential in Ch(Y*, Z*) is determined by the formula D({fn}) = {dZfn - (* *-1)pfn-1dY }, for fn 2 hom(Yn, Zn+p). 4.2. Enriched categories. A category O enriched over C, or a C-category, consis* *ts of a class I (representing the objects of O), and, for any objects i, j, k 2 I, a C-object O* *(i, j) (representing the morphisms from i to j in O) and C-morphisms O(i, j) O(j, k) -! O(i, k), and 1 ! O(i, i) (representing the composition of morphisms in O and the identity morphism on i)* *. These struc- ture morphisms are required to be associative and unital in the evident sense. * *Notice that a closed symmetric monoidal cateory C is enriched over itself since C(Y, Z) is an object* * of C. Following [20], we use the term CI-category to signify a category enriched over C, whose * *set of objects is I. Let O be a CI-category and R be a category enriched over C. A (covariant) fun* *ctor enriched over C, or C-functor from O to R, M :O -! R, consists of an R-object M(i) for every i 2 I, and of morphisms in C M(i, j): O(i, j) -! R(M(i), M(j)), for every i, j 2 I, that are associative and unital. There is an analogous noti* *on of a contravariant C-functor. A natural transformation enriched over C, : M ! M0, between two C-functors M* *, M0:O ! R consists of C-morphisms i:1 -! R(M(i), M0(i)) for every object i of O, that satisfy the obvious commutativity conditions for * *a natural trans- formation (see [5, 6.2.4]). Notice that if R = C then a morphism i:1 ! C(M(i),* * M0(i)) is the same as the adjoint morphism (i): M(i) ! M0(i) in C. For fixed C and I, we consider the collection of CI-categories as a category * *in its own right. A morphism of CI-categories is an enriched functor that is the identity on the se* *t of objects. Suppose now that C is a monoidal model category. In particular, C is equipped* * with a notion of weak equivalence. Then we say that a morphism : O ! R of CI-categories is a w* *eak equivalence CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 11 if it is a weak equivalence pointwise, i.e., if the map O(i, j) ! R(i, j) is a * *weak equivalence in C for all i, j 2 I. 4.3. Homotopy theory of right modules over enriched categories. For a CI-catego* *ry O, a (right) O-module is a contravariant C-functor from O to C. Explicitly an O-mo* *dule M consists of objects M(i) in C for i 2 I and (since C is a closed monoidal category and s* *ince it is enriched over itself) of C-morphisms M(j) O(i, j) -! M(i) which are associative and unital. A morphism of O-modules, : M ! M0, is an e* *nriched natural transformation, i.e., a collection of C-morphism (i): M(i) ! M0(i) sat* *isfying the usual naturality requirements. Such a morphism of O-module is a weak equivalence if * *each (i) is a weak equivalence in C. We denote by Mod -O the category of right O-modules an* *d natural transformations. Let : O ! R be a morphism of CI-categories. Clearly, induces a restrictio* *n of scalars functor on module categories * : Mod -R -! Mod -O M 7-! M O . As explained in [20, page 323], the functor * has a left adjoint functor *, a* *lso denoted - O R (one can think of * as the left Kan extension). Schwede and Shipley [20, Theor* *em 6.1] prove that under some technical hypotheses on C, the category Mod -O has a Quillen mo* *dule structure, and moreover, if is a weak equivalence of CI-categories, then the pair ( *, * **) induces a Quillen equivalence of module categories. We will need this result in the case C = Ch. In keeping with our notation, we* * use ChI-categories to denote categories enriched over chain complexes, with object set I. Note tha* *t the category of modules over a ChI-categories admits coproducts (i.e. direct sums). Theorem 4.1 (Schwede-Shipley, [20]). (1) Let O be a ChI-category. Then Mod -O has a cofibrantly generated Quillen * *model structure, with fibrations and weak equivalences defined objectwise. (2) Let : O ! R be a weak equivalence of ChI-categories. Then ( *, *) indu* *ce a Quillen equivalence of the associated module categories. Proof.General conditions on C that guarantee the result are given in [20, Theor* *em 6.1]. It is straightforward to check that the conditions are satisfied by the category of c* *hain complexes (the authors of [20] verify them for various categories of spectra, and the ver* *ification for chain complexes is strictly easier). Let O and R be CI-categories and let M and N be right modules over O and R re* *spectively. A morphism of pairs (O, M) ! (R, N) consists of a morphism of CI-categories : * *O ! R and a morphism of O-modules : M ! *(N). The corresponding category of pairs (O, M) * *is called the CI-module category. A morphism ( , ) in CI-module is called a weak equival* *ence if both and are weak equivalences. Two objects of CI-module are called weakly equival* *ent if they are linked by a chain of weak equivalences, pointing in either direction. In our study of the formality of the little balls operad, we will consider ce* *rtain splittings of O-modules into direct sums. The following homotopy invariance property of such * *a splitting will be important. 12 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C Proposition 4.2. Let (O, M) and (O0, M0) be weakly equivalent ChI-modules. If M* * is weakly equivalent as an O-module to a direct sum Mn, then M0 is weakly equivalent as * *an O0-module to a direct sum M0nsuch that (O, Mn) is weakly equivalent to (O0, M0n) for eac* *h n. Proof.It is enough to prove that for a direct weak equivalence ( , ) : (O, M) -'!(R, N), M splits as a direct sum if and only if N splits in a compatible way. In one direction, suppose that N ' nNn as R-modules. It is clear that the r* *estriction of scalars functor * preserves direct sums and weak equivalences (quasi-isomorphi* *sms). Therefore *(N) ' n *(Nn). Since by hypothesis M is weakly equivalent to *(N), we have * *the required splitting of M. In the other direction suppose that the O-module M is weakly equivalent to n* *Mn. We can assume that each Mn is cofibrant, hence so is nMn. Moreover *(N) is fibrant b* *ecause every O-module is. Therefore, since M is weakly equivalent to *(N), there exists a * *direct weak equivalence fl : n Mn '! *(N). Since ( *, *) is a Quillen equivalence, the wea* *k equivalence fl induces an adjoint weak equivalence fl[: *( nMn) '!N. As a left adjoint, * com* *mutes with coproducts, therefore we get the splitting n *(Mn) '!N. Moreover we have a wea* *k equivalence Mn '! * *(Mn) because it is the adjoint of the identity map on *(Mn), Mn is co* *fibrant, and ( *, *) is a Quillen equivalence. Thus that splitting of N is compatible with * *the given splitting of M. 4.4. Lax monoidal functors, enriched categories, and their modules. Let C and D* * be two symmetric monoidal categories. A lax symmetric monoidal functor F :C ! D is a (* *non enriched) functor, together with morphisms 1D ! F (1C) and F (X) F (Y ) ! F (X Y ), n* *atural in X, Y 2 C, that satisfy the obvious unit, associativity, and symmetry relations.* * In this paper, we will some times use "monoidal" to mean "lax symmetric monoidal", as this is the* * only notion of monoidality that we will consider. Such a lax symmetric monoidal functor F induces a functor (which we will stil* *l denote by F ) from CI-categories to DI-categories. Explicitly if O is a CI-category then F (O* *) is the D-category whose set of objects is I and morphisms are (F (O))(i, j) := F (O(i, j)). Moreo* *ver, F induces a functor from Mod -O to Mod -F (O). We will denote this functor by F as well. The main examples that we will consider are those from Sections 2.0.1 and 2.0* *.2, and their composites: (1) Homology: H *:(Ch , , K) -! (Ch , , K); (2) Normalized singular chains: C* :(Top, x, *) -! (Ch , , K) , X 7-! C*(X). The fact that the normalized chains functor is lax monoidal, and equivalent to * *the unnormalized chains functor, is explained in [20, Section 2]. As is customary, we often abbr* *eviate the composite H *O C*as H*. Recall that we also use the functor Hn :(Ch , , K) ! (Ch , , K), where Hn(C* *, d) is seen as a chain complex concentrated in degree n. The functor Hn is not monoidal for n > * *0. However, H0 is monoidal. Thus if B is a small TopI-category then C*(B) and H*(B) are ChI-categories. A* *lso if B :B ! Top is a B-module then C*(B) is a C*(B)-module and H*(B) is an H*(B)-module. We* * also have the ChI-category H0(B). CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 13 4.5. Discretization of enriched categories. When we want to emphasize that a ca* *tegory is not enriched (or, equivalently, enriched over Set), we will use the term discre* *te category. When we speak of an A-diagram we always assume that A is a discrete category. Let C be a closed symmetric monoidal category. There is a forgetful functor * *OE: C ! Set, defined by OE(C) := homC(1, C) It is immediate from the definitions that OE is a monoidal functor. Therefore, * *it induces a functor from categories enriched over C to discrete categories. We will call this indu* *ced functor the discretization functor. Let O be a category enriched over C. The discretizati* *on of O will be denoted Offi. It has the same objects as O, and its sets of morphisms are given* * by the discretization of morphisms in O. For example, Top can be either the Top-enriched category or * *the associated discrete category. For Ch, the set of morphisms between two chain complexes X* * *and Y* in the discretization of Ch is the set of cycles of degree 0 in the chain complex Ch(X* **, Y*), i.e. the set of chain maps. It is easy to see that if C is a closed symmetric monoidal c* *ategory, then the discretization of C is the same as C, considered as a discrete category. We wil* *l not use special notation to distinguish between C and its underlying discrete category. Let M :O ! R be a C-functor between two C-categories. The underlying discrete* * functor is the functor Mffi:Offi-! Rffi induced in the obvious way from M. More precisely, if i is an object of O then * *Mffi(i) = M(i). If j is another object and f 2 Offi(i, j), that is f :1 ! O(i, j), then Mffi(f)* * 2 Rffi(Mffi(i), Mffi(j)) M(i,j) * * 0 is defined as the composite 1 !f O(i, j) -! R(M(i), M(j)). Similarly if : M* * ! M is an enriched natural transformation between enriched functors, we have an induced d* *iscrete natural transformation ffi:Mffi! M0ffi. In particular, an O-module M induces an Offi-d* *iagram Mffiin C and a morphism of O-modules induces a morphism of Offi-diagrams. Let F :C ! D be a lax symmetric monoidal functor, let O be a CI-category, and* * let M :O ! C be an O-module. As explained before, we have an induced DI-category F (O), and* * an F (O)- module F (M). We may compare Offiand F (O)ffiby means of a functor FOffi:Offi-! F (O)ffi which is the identity on objects and if f :1C ! O(i, j) is a morphism in Offi, * *then FOffi(f) is the F(f) composite 1D ! F (1C) ! F (O(i, j)). It is straightforward to verify the following two properties of discretizatio* *n. Lemma 4.3. Let F :C ! D be a lax symmetric monoidal functor, let O be a CI-cate* *gory and let M be an O-module. The following diagram of discrete functors commutes __Mffi//_ Offi C FffiO|| |F| fflffl| fflffl| F (O)ffiF(M)ffi//_D. Lemma 4.4. Let C be a monoidal model category and let O be a CI-category. If :* * M -'! M0 is a weak equivalence of O-modules then ffi:Mffi'-!M0ffiis a weak equivalence * *of Offi-diagrams. 14 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C 5. Operads and associated enriched categories We will first recall the notions of operads, right modules over operads, and * *weak equivalences of operads. We will then describe the enriched category associated to an operad* *. Finally, we will treat the central example of the little balls operad. The enriched category vie* *wpoint will help us to deduce (in Section 7) the formality of certain topological functors from the* * formality of the little ball operads. 5.1. Operads and right modules. Among the many references for operads, a recent* * one that covers them from a viewpoint similar to ours is Ching's paper [6]. However, the* *re is one important difference between our setting and Ching's: He only considers operads without t* *he zero-th term, while we consider operads with one. Briefly, an operad in a symmetric monoidal * *category (C, , 1), or a C-operad, is a symmetric sequence O(o) = {O(n)}n2N of objects of C, equipp* *ed with structure maps O(n) O(m1) . . .O(mn) -! O(m1+ . .+.mn) and 1 -! O(1), satisfying certain associativity, unit, and symmetry axioms. There is an obvio* *us notion of a morphism of operads. When C is a monoidal model category, we say that a morphism f :O(o) ! R(o) of* * C-operads is a weak equivalence if f(n) is a weak equivalence in C for each natural number n. * *If f :O(o) ! R(o) and f0:O0(o) ! R0(o) are morphisms of operads, a morphism of arrows from f to f* *0 is a pair (o: O(o) ! O0(o) , r :R(o) ! R0(o))of morphisms of operads such that the o* *bvious square diagrams commute. Such a pair (o, r) is called a weak equivalence if both o an* *d r are weak equivalences. A right module over a C-operad O(o) is a symmetric sequence M(o) = {M(n)}n2N * *of objects of C, equipped with structure morphisms M(n) O(m1) . . .O(mn) -! M(m1+ . .+.mn) satisfying certain obvious associativity, unit, and symmetry axioms (see [6] fo* *r details). Notice that a morphism of operads f :O(o) ! R(o) endows R(o) with the structure of a r* *ight O(o)- module. 5.2. Enriched category associated to an operad. Fix a closed symmetric monoidal* * category C that admits finite coproducts. Recall from Section 4.2 that a CN-category is * *a category enriched over C whose set of objects is N. The CN-category associated to the C-operad O(* *o) is the category O defined by a O(m, n) = O(ff-1(1)) . . .O(ff-1(n)) ff:m_-!n_ where the coproduct is taken over set maps ff: m_ := {1, . .,.m} ! n_ := {1, . * *.,.n} and O(ff-1(j)) = O(mj) where mj is the cardinality of ff-1(j). Composition of morp* *hisms is pre- scribed by operad structure maps in O(o). In particular O(m, 1) = O(m). Let O(o) be a C-operad and let O be the associated CN-category. A right modul* *e (in the sense of operads) M(o) over O(o) gives rise to a right O-module (in the sense of Sect* *ion 4) M(-): O - ! C n7-! M(n) CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 15 where M(-) is defined on morphisms by the C-morphisms M(m, n): O(m, n) -! C(M(n), M(m)) obtained by adjunction from the structure maps a M(n) O(m, n) = M(n) O(ff-1(1)) . . .O(ff-1(n)) -! M(m). ff:m_-!n_ If f :O(o) ! R(o) is a morphism of operads then we have an associated right O-m* *odule R(-): O ! C. It is obvious that if O(o) and O0(o) are weakly equivalent, objectwise cofibr* *ant, operads over a monoidal model category C then the associated CN-categories O and O0are weakl* *y equivalent. Also, if f :O(o) ! R(o) and f0:O0(o) ! R0(o) are weakly equivalent morphisms of* * operads, then the pair (O, R(-)) is weakly equivalent, in the category of CN-modules, to the * *pair (O0, R0(-)). Let F :C ! D be a lax symmetric monoidal functor, and suppose O(o) is an oper* *ad in C. Let O be the CN-category associated to O(o). Then F (O(o)) is an operad in D, a* *nd F (O) is a DN-category. It is easy to see that there is a natural morphism from the DN-cat* *egory associated to the D-operad F (O(o)) to F (O). This morphism is not an isomorphism, unless * *F is strictly monoidal and also takes coproducts to coproducts, but in all cases that we cons* *ider, it will be a weak equivalence. Similarly if f :O(o) ! R(o) is a morphism of operads and if* * R(-) is the right O-module associated to the O(o)-module R(o), then F (R(-)) has a natural * *structure of an F (O)-module, extending the structure of an F (O(o))-module possessed by F (R(o* *)). 5.3. The standard little balls operad. The most important operad for our purpos* *es is what we will call the standard balls operad. Let V be a Euclidean space. By a standa* *rd ball in V we mean a subset of V that is obtained from the open unit ball by dilation and tra* *nslation. The operad of standard balls will be denoted by B(o, V ). It is the well-known oper* *ad in (Top, x, *), consisting of the topological spaces B (n, V ) = {n-tuples of disjoint standard balls inside the unitVball* *}of with the structure maps given by composition of inclusions after suitable dilat* *ions and transla- tions. The TopN-category associated to the standard balls operad B(o, V ) will be de* *noted by B(V ). An object of B(V ) is a non-negative integer n which can be thought of as an ab* *stract (i.e., not embedded) disjoint union of n copies of the unit ball in V . The space of morph* *isms B(V )(m, n) is the space of embeddings of m unit balls into n unit balls, that on each ball* * are obtained by dilations and translations. Let j :W ,! V be a linear isometric inclusion of Euclidean spaces. Such a ma* *p induces a morphism of operads j :B(o, W ) -! B(o, V ) where a ball centered w 2 W is sent to the ball of same radius centered at j(w). Hence B(o, V ) is a right module over B(o, W ), and we get a right B(W )-modu* *le B (-, V ): B(W-)! Top n 7-! B(n, V ). 16 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C We can apply lax monoidal functors to the above setting. For example, C *(B * *(o, W )) and H *(B (o, W )) are operads in (Ch , , K). Hence we get ChN-categories C*(B(W )* *) and H*(B(W )), a right C*(B(W ))-module C*(B (-, V )), and a right H*(B(W ))-module H*(B (-, V* * )). We will also consider the discrete categories B(W )ffiand C*(B(W ))ffiobtaine* *d by by the dis- cretization process from B(W ) and C*(B(W )) respectively. Note that C*(B(W ))f* *fi= K[B(W )ffi]. 6.Formality and splitting of the little balls operad In this section, all chain complexes and homology groups are taken with coeff* *icients in R. A deep theorem of Kontsevich (Theorem 1.2 of the Introduction and Theorem 2 of [1* *4]) asserts that the standard balls operad is formal over the reals. We will need a slight * *strengthening of this result. Throughout this section, let j :W ,! V be, as usual, a linear iso* *metric inclusion of Euclidean spaces. Recall the little balls operad and the associated enriche* *d categories and modules as in Section 5.3. Here is the version of Kontsevich's theorem we need. Theorem 6.1 (Relative Formality). If dimV > 2 dimW then the morphism of chain o* *perads C *(j): C*(B (o, W )) R -! C*(B (o, V )) R is weakly equivalent to the morphism H *(j): H*(B (o, W ); R) -! H*(B (o, V ); R). Sketch of the proof.A detailled proof will appear in [16]. Here we give a sket* *ch based on the proof absolute formality given in [14, Theorem 2], and we follow that paper's n* *otation. Denote by FM d(n) the Fulton-MacPherson compactification of the configuration space of* * n points in Rd. This defines an operad FM d(o) which is homotopy equivalent to the little balls* * operad B(o, Rd). Kontsevich constructs a quasi-isomorphism : SemiAlgChain*(FM d(n)) -'!Graphsd(n) ^R where SemiAlgChain*is a chain complex of semi-algebraic chains naturally quasi-* *isomorphic to singular chains and Graphsdis the chain complex of admissible graphs defined in* * [14, Definition 13]. For , a semi-algebraic chain on FM d(n), the map is defined by X (,) = , where the sum is taken over all admissible graphs and ! is the differential * *form defined in [14, Definition 14]. Let j*: FM dimW(n) ! FM dimV(n) be the map induced by the inclusion of Euclid* *ean spaces j. Notice that Hi(j*) = 0 for i > 0. Define ffl: GraphsdimW(n) ! GraphsdimV(n) * *to be zero on graphs with at least one edge, and the identity on the graph without edges. We * *need to show that the following diagram commutes: SemiAlgChain*(FM dimW (n))'oGraphsdimW(n)o^R__'__//H*(FM dimW (n)) |j*| |ffl| |H(j*)| fflffl| ' fflffl| ' fflffl| SemiAlgChain*(FM dimV(n))oo__GraphsdimV(n) ^R____//H*(FM dimV(n)). CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 17 The commutativity of the right hand square is clear. For the left hand square i* *t suffices to check that for any admissible graph of positive degree and for any non-zero semi-al* *gebraic chain , 2 SemiAlgChain*(FM dimW (n)) we have = 0. The first n vertices of , 1, . .,.n, are called external and the other are c* *alled internal. If every external vertex of is connected to an edge, then, using the fact that interna* *l vertices are at least trivalent, we obtain that the form ! on FM dimV(n) is of degree n(dim * *V - 1)=2. Since dim V > 2 dimW , we get that deg(! ) > dim(FM dimW (n)). Therefore deg(! ) > de* *g(j*(,)) and = 0. If has an isolated external vertex, then = , where * *,0 is a chain in SemiAlgChain*(FM dimW (m)) with m < n and the proof proceeds by induction. We remark once again that the formality theorem is for chain complexes over R* *, not over Q. We do not know if the little balls operad is formal over the rational numbers, * *but we do think it is an interesting question. We note that a general result about descent of f* *ormality from R to Q was proved in [12], for operads without a term in degree zero. The proof does* * not seem to be easily adaptable to operads with a zero term. To deduce the formality of certain diagrams more directly related to spaces o* *f embeddings, we first reformulate relative formality in terms of homogeneous splittings in the * *spirit of Proposi- tion 3.3. With this in mind we introduce the following enrichment of Definition* * 3.2. Definition 6.2. Let O be a Ch I-category. We say that an O-module M :O ! Ch s* *plits homogeneously if there exists a sequence {Mn}n2N of O-modules such that M ' nM* *n and H *(Mn) = Hn(Mn). Our first example (a trivial one) of such a homogeneous splitting of modules * *is given by the following Lemma 6.3. If dimV > dimW then the H*(B(W ))-module H*(B (-, V )) splits homoge* *nously. Proof.Notice that H0(B(W )) is also a ChN-category and we have an obvious inclu* *sion functor (because our chain complexes are non-negatively graded) i: H0(B(W )) ,! H*(B(W )) and a projection functor (because our chain complexes have no differentials) : H*(B(W )) -! H0(B(W )) between Ch N-categories, where O i is the identity. Therefore, an H *(B(W ))* *-module admits a structure of an H 0(B(W ))-module via i. Since H 0(B(W )) is a category of c* *hain complexes concentrated in degree 0 and H*(B (-, V )) has no differentials, it is clear th* *at we have a splitting of H0(B(W ))-modules (3) H*(B (-, V )) ~= 1n=0Hn(B (-, V )). Moreover, since dimW < dimV the morphisms H*(B (n, W )) -! H*(B (n, V )) are zero in positive degrees. Hence the H *(B(W ))-module structure on H *(B (* *-, V )) factors through the above-mentioned H0(B(W ))-module structure via . Therefore, the sp* *litting (3) is a splitting of H*(B(W ))-modules. 18 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C Using Lemma 6.3 and the Relative Formality Theorem, we obtain the following h* *ighly non- trivial splitting. Lemma 6.4. If dimV > 2 dimW then the C*(B(W ))-module C*(B (-, V )) splits homo* *genously. Proof.We deduce from Theorem 6.1 that the ChN-module categories (C *(B(W )), C** *(B (-, V ))) and (H *(B(W )), H*(B (-, V ))) are equivalent. By Lemma 6.3 the latter splits* * homogeneously, hence, by Proposition 4.2 the same is true of the former. Recall from Section 4.5 that the enriched category B(W ) has an underlying di* *screte category B(W )ffiand that the B(W )-module B(-, V ) induces a B(W )ffi-diagram B(-, V )f* *fi. Proposition 6.5. If dimV > 2 dimW then the B(W )ffi-diagram C *(B (-, V ))ffi:B(W )ffi-! ChR is formal. Proof.By Lemma 4.3 the following diagram of discrete functors commutes: B(-,V )ffi B(W )ffi__________//_Top (C*)ffiB(W)|| |C*| fflffl|(C*(B(V,-)fflffl|))ffi (C*(B(W )))ffi_______//_ChR We want to prove that the B(W )ffi-diagram C* B (-, V )ffiis formal. By the com* *mutativity of the square above and Lemma 3.6 it is enough to prove that the (C *(B(W )))ffi-diagr* *am (C *(B (V, -)))ffi is formal. By Lemma 6.4 the C*(B(W ))-module C*(B (-, V )) splits homogeneously* *. By Lemma 4.4 we deduce that the C*(B(W ))ffi-diagram C*(B (-, V ))ffisplits homogeneousl* *y, which implies by Proposition 3.3 the formality of that diagram. 7.Formality of a certain diagram arising from embedding calculus In this section, all chain complexes are still taken over the real numbers. A* *s before, fix a linear isometric inclusion of Euclidean vector spaces j :W ,! V . Let O(W ) be the pos* *et of open subsets of W . As explained in the Introduction, we have two contravariant functors Emb (-, V ) , Imm (-, V ): O(W ) -! Top. Moreover, the fixed embedding j :W ,! V can serve as a basepoint, so we can co* *nsider the homotopy fiber of the inclusion Emb (-, V ) ! Imm (-, V ), which we denote by ____ Emb (-, V ): O(W ) -! Top. Our goal in this section is to compare a certain variation of this functor wi* *th the functor B (-, V )ffi:B(W )ffi-! Top and to deduce in Theorem 7.2 the stable formality of certain diagrams of embedd* *ing spaces. In order to do this we first introduce a subcategory Os(W ) of O(W ) and a categor* *y fOs(W ) which will serve as a turning table between Os(W ) and B(W )ffi. CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 19 To describe Os(W ) recall that a standard ball in W is an open ball in the me* *tric space W , i.e. it is obtained in a unique way by a dilation and translation of the unit ball i* *n W . The category Os(W ) is the full subcategory of O(W ) whose objects are finite unions of disj* *oint standard balls. The category fOs(W ) is a kind of covering of Os(W ). Recall that the object * *m 2 N of B(W ) can be thought of as an abstract disjoint union of m copies of the unit ball of* * W . An object of fOs(W ) is then an embedding OE: m ,! W such that the restriction of OE to each* * unit ball amounts to a dilation and translation. In other words an object (OE, m) of fOs(W ) is t* *he same as an ordered m-tuple of disjoint standard balls in W . The union of these m standard balls * *is an object of Os(W ) that we denote by OE(m), as the image of the embedding OE. By definitio* *n, there is a morphism in fOs(W ) between two objects (OE, m) and (_, n) if and only if OE(n)* * _(m), and such a morphism is unique. We define functors B(W )ffi~oofOs(W_)_ss//_Os(W ). Here ss is defined on objects by ss(OE, m) = OE(m) and is defined on morphisms * *by sending a morphism ff : (OE1, m1) ! (OE2, m) to the inclusion OE1(m1) ,! OE2(m2). The fun* *ctor ~ is defined on objects by ~(OE, m) = m, and is defined on morphisms using the fact that any tw* *o standard balls in W can be canonically identified by a unique transformation that is a combina* *tion of dilation and translation. We would like to compare the following two composed functors ____ ____ ss s Emb(-,V ) Emb (ss(-), V )): fOs(W_)//_O (W_)_______//Top _~__//_ ffiB_(-,V_)ffi//_ B(~(-), V )ffi:fOs(W )B(W ) Top. ____ Proposition 7.1. The fOs(W )-diagrams B(~(-), V )ffiand Emb (ss(-), V ) are wea* *kly equivalent. Proof.Define subspaces AffEmb(OE(n), V ) Emb (OE(n), V ) and AffImm(OE(n), V * *) Imm (OE(n), V ) to be the spaces of embeddings and immersions, respectively, that are affine on* *_each_ball. It is well- known that the above inclusion maps are homotopy equivalences. We may define Af* *fEmb(OE(n), V ) to be the homotopy fiber of the map AffEmb(OE(n), V ) ! AffImm(OE(n), V ). Thus* * there is a natural homotopy equivalence _______ ____ AffEmb (OE(n), V ) -'!Emb (OE(n), V ). Define Inj(W, V ) as the space of injective linear maps from W to V , quotiente* *d out by the multiplicative group of positive reals, i.e. defined up to scaling. Then there * *is a natural homotopy equivalence AffImm(OE(n), V ) -'!Inj(W, V )n obtained by differentiating the immersion at each component of OE(n). Moreover * *the map AffEmb(OE(n), V ) -! Inj(W, V )n is a fibration and we denote its fiber by F (n, OE). So we get a natural equiva* *lence _______ AffEmb(OE(n), V ) -! F (n, OE). Finally since the composite map B(n, V ) ,! AffEmb(OE(n), V ) -! Inj(W, V )n 20 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C is the constant map into the basepoint, there is a natural map B(n, V ) ! F (n,* * OE). It is easy to see that the map is an equivalence. To summarize, we have constructed the follo* *wing chain of natural weak equivalences ____ ' _______ ' ' Emb (OE(n), V ) AffEmb (OE(n), V ) -! F (n, OE) B (n, V ). We are ready to prove the main result of this section. ____ Theorem 7.2. If dimV > 2 dimW then the fOs(W )-diagram C*(Emb (ss(-), V )) is s* *tably formal. Proof.By Proposition 6.5 and Lemma 3.6 the diagram C*(B (~(-), V ))ffiis stably* * formal. Propo- sition 7.1 implies the theorem. 8.More generalities on calculus of functors In this section we digress to review in a little more detail the basics of em* *bedding and orthogonal calculus. We will also record some general observations about bi-functors to wh* *ich both brands of calculus apply. The standard references are [24] and [23]. 8.1. Embedding calculus. Let M be a smooth manifold (for convenience, we assume* * that M is the interior of a compact manifold with boundary). Let O(M) be the poset of * *open subsets of M and let Ok(M) be the subposet consisting of open subsets homeomorphic to a* * union of at most k open balls. Embedding calculus is concerned with the study of contravar* *iant functors (cofunctors) from F to a Quillen model category (Weiss only considers functors * *into the category of spaces, and, implicitly, spectra, but much of the theory works just as well * *in the more general setting of model categories). Following [24, page 5], we say that a cofunctor i* *s good if it converts isotopy equivalences to weak equivalences and filtered unions to homotopy limit* *s. Polynomial cofunctors are defined in terms of certain cubical diagrams, similarly to the w* *ay they are defined in Goodwillie's homotopy calculus. Recall that a cubical diagram of spaces is * *called strongly co-cartesian if each of its two-dimensional faces is a homotopy pushout square.* * A cofunctor F on O(M) is called polynomial of degree k if it takes strongly co-cartesian k + 1-d* *imensional cubical diagrams of opens subsets of M to homotopy cartesian cubical diagrams (homotopy* * cartesian cubical diagrams is synonymous with homotopy pullback cubical diagrams). Good * *cofunctors can be approximated by the stages of the tower defined by TkF (U) = holim F (U0). {U02Ok(M)|U0 U} It turns out that TkF is polynomial of degree k, and moreover there is a natu* *ral map F -! TkF which in some sense is the best possible approximation of F by a polynomial fun* *ctor of degree k. More precisely, the map F -! TkF can be characterized as the essentially unique* * map from F to a polynomial functor of degree k that induces a weak equivalence when evalua* *ted on an object of Ok(M). In the terminology of [24], TkF is the k-th Taylor polynomial of F . * *F is said to be homogeneous of degree k if it is polynomial of degree k and Tk-1F is equivalent* * to the trivial functor. For each k, there is a natural map TkF ! Tk-1F , compatible with the m* *aps F ! TkF and F ! Tk-1F . Its homotopy fiber is a homogeneous functor of degree k, and it* * is called the k-th layer of the tower. It plays the role of the k-th term in the Taylor serie* *s of a function. For space-valued functors, there is a useful general formula for the k-th layer in * *terms of spaces of CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 21 M M sections of a certain bundle p : E ! k over the space k of unordered k-tupl* *es of distinct points in M. The fiber of p at a point m_= {m1, . .,.mk} is "F (m), which is de* *fined to be the total fiber of the k-dimensional cube S 7! F (N(S)) where S ranges over subsets* * of m_and N(S) stands for a "small tubular neighborhood" of S in M, i.e., a disjoint union of * *open balls in M. The fibration p has a preferred section. See [24], especially Sections 8 and 9,* * for more details and a proof of the following proposition. Proposition 8.1 (Weiss). The homotopy fiber of the map TkF ! Tk-1F is equivalen* *t to the space of sections of the fibration p above which agree with the preferred secti* *on in a neighborhood of the fat diagonal in Mk. We denote this space of restricted sections by ` ` ' ' M c , [F (k). k Even though TkF is defined as the homotopy limit of an infinite category, for* * most moral and practical purposes it behaves as if it was the homotopy limit of a very small c* *ategory (i.e., a category whose simplicial nerve has finitely many non-degenerate simplices). Th* *is is so because of the following proposition. Proposition 8.2. There is a very small subcategory C of Ok(M) such that restric* *tion from Ok(M) to C induces an equivalence on homotopy limits of all good cofunctors. Proof.It is not difficult to show, using handlebody decomposition and induction* * (the argument is essentially contained in the proof of Theorem 5.1 of [24]) that one can find* * a finite collection {U1, . .,.UN } of open subsets of M such that all their possible intersections * *are objects of Ok(M) and Mk = [Ni=1Uki This is equivalent to saying that the sets Ui cover M in what Weiss calls the G* *rothendieck topology Jk. By [24], Theorem 5.2, polynomial cofunctors of degree k are homoto* *py sheaves with respect to Jk. In practice, this means the following. Let C be the subposet of * *Ok(M) given by the sets Ui and all their possible intersections (clearly, C is a very small ca* *tegory). Let G be a polynomial cofunctor of degree k. Then the following canonical map is a homotop* *y equivalence G(M) -! holimG(U). U2C We conclude that for a good cofunctor F , there is the following zig-zag of w* *eak equivalences. holimF (U) -'!holimTkF (U) -' TkF (M) U2C U2C Here the left map is a weak equivalence because the map F ! TkF is a weak equiv* *alence on objects of Ok(M), and all objects of C are objects of Ok(M). The right map is a* *n equivalence because TkF is a polynomial functor of degree k, in view of the discussion abov* *e. The important consequence of the proposition is that TkF commutes, up to a zi* *g-zag of weak equivalences, with filtered homotopy colimits of functors. In the same spirit, * *we have the following proposition. 22 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C Proposition 8.3. Let F : Ok(M) ! ChQ be a good cofunctor into rational chain c* *omplexes. Then the natural map (TkF (M)) R -! Tk(F R)(M) is a weak equivalence. Proof.Tensoring with R obviously commutes up to homotopy with very small homoto* *py limits, and so the claim follows from Proposition 8.2. 8.2. Orthogonal calculus. The basic reference for Orthogonal calculus is [23]. * *Let J be the topological category of Euclidean spaces and linear isometric inclusions. Ortho* *gonal calculus is concerned with the study of continuous functors from J to a model category enri* *ched over Top. We will only consider functors into Top, Spectraand closely related categories.* * Like embedding calculus, orthogonal calculus comes equipped with a notion of a polynomial func* *tor, and with a construction that associates with a functor G a tower of approximating functo* *rs PnG such that PnG is, in a suitable sense, the best possible approximation of G by a pol* *ynomial functor of degree n. Pn is defined as a certain filtered homotopy colimit of compact h* *omotopy limits. For each n, there is a natural map PnG ! Pn-1G and its fiber (again called the * *n-th layer) is denoted by DnG. DnG is a homogeneous functor, in the sense that it is polynomia* *l of degree n and Pn-1DnG ' *. The following characterization of homogeneous functors is prov* *ed in [23]. Theorem 8.4 (Weiss). Every homogeneous functor of degree n from vector spaces t* *o spectra is equivalent to a functor of the form Cn ^ SnV hO (n) where Cn is a spectrum with an action of the orthogonal group O(n), SnV is the * *one-point com- pactification of the vector space Rn V , and the subscript h O(n) denotes hom* *otopy orbits. It follows, in particular, that given a (spectrum-valued) functor G to which * *orthogonal calculus applies, DnG has the form described in the theorem, with some spectrum Cn. The * *spectrum Cn is called the n-th derivative of G. There is a useful description of the derivativ* *es of G as stabilizations of certain types of iterated cross-effects of G. Let G1, G2 be two functors to which orthogonal calculus applies. Let ff : G1 * *! G2 be a natural transformation. Very much in the spirit of Goodwillie's homotopy calculus, we s* *ay that G1 and G2 agree to n-th order via ff if the map ff(V ) : G1(V ) ! G2(V ) is (n + 1) di* *m(V ) + c-connected, where c is a possibly negative constant, independent of V . Using the descripti* *on of derivatives in terms of cross-effects, it is easy to prove the following proposition Proposition 8.5. Suppose that G1 and G2 agree to n-th order via a natural trans* *formation ff: G1 ! G2. Then ff induces an equivalence on the first n derivatives, and the* *refore an equiva- lence on n-th Taylor polynomials Pnff : PnG1 -'!PnG2 8.3. Bifunctors. In this paper we consider bifunctors E :O(M)opx J -! Top= Spectra such that the adjoint cofunctor O(M) ! Funct(J , Top= Spectra) is good (in the * *evident sense) and the adjoint functor J ! Funct(O(M)op, Top= Spectra) is continuous. We may a* *pply both embedding calculus and orthogonal calculus to such a bifunctor. Thus by PnE(M, * *V ) we mean CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 23 the functor obtained from E by considering it a functor of V , (with M being a * *"parameter") and taking the n-th Taylor polynomial in the orthogonal sense. Similarly, TkE(M, V * *) is the functor obtained by taking the k-th Taylor polynomial in the sense of embedding calculu* *s. We will need a result about the interchangeability of order of applying the d* *ifferential operators Pn and Tk. Operator Tk is constructed using a homotopy limit, while Pn is const* *ructed using a homotopy limit (over a compact topological category) and a filtered homotopy co* *limit. It follows that there is a natural transformation PnTkE(M, V ) -! TkPnE(M, V ) and a similar natural transformation where Pn is replaced with Dn. Lemma 8.6. Let E be a bifunctor as above. For all n and k the natural map PnTkE(M, V ) -'!TkPnE(M, V ) is an equivalence. There is a similar equivalence where Pn is replaced by Dn. Proof.By Proposition 8.2, Tk can be presented as a very small homotopy limit. * *Therefore, it commutes up to homotopy with homotopy limits and filtered homotopy colimits. Pn* * is constructed using homotopy limits and filtered homotopy colimits. Therefore, Tk and Pn comm* *ute. 9. Formality and the embedding tower In this section we assume that ff: M ,! W is an inclusion of an open subset i* *nto a Euclidean space W . From our point of view, there is no loss of generality in this assump* *tion, because if M is an embedded manifold in W ,_we_can replace M with an open tubular neighborho* *od, without changing the homotopy type of Emb (M, V ). As usual, we fix an isometric inclus* *ion j :W ,! V of Euclidean vector spaces. Recall that we defined the functor ____ Emb (-, V ): O(M) -! Top. ____ ____ The stable rationalisation HQ ^ Emb (-, V )+ of Emb (-, V ) admits a Taylor t* *ower (in this section, Taylor towers are taken in the sense of embedding calculus). Our goal* * is to give in Theorem 9.3 a splitting of the k-th stage of this tower. The splitting is not a* *s a product of the layers in the embedding towers. Rather, we will see in the next section that th* *e splitting is as a product of the layers in the orthogonal tower. Recall the poset Os(W ) of finite unions of standard balls in W from Section * *7. Let Os(M) be the full subcategory of Os(W ) consisting of the objects which are subsets of M* *. For a natural number k we define Osk(M) as the full subcategory of Os(M) consisting of disjoi* *nt unions of at most k standard balls in M. Proposition 9.1. Let M be an open submanifold of a vector space W and let F :O(* *M) ! Top be a good functor. The restriction map TkF (M) := holim F (U) -! holim F (U), U2Ok(M) U2Osk(M) induced by the inclusion of categories Osk(M) ! Ok(M), is a homotopy equivalenc* *e. Proof.Define TksF (M) := holim F (U). There are projection maps U2Osk(M) TksF (M) -! Tks-1F (M) 24 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C induced by the inclusion of categories Osk-1(M) ! Osk(M), and the map TkF ! Tks* *F extends to a map of towers. One can adapt the methods of [24] to analyze the functors TksF* * . In particular, it is not hard to show, using the same methods as in [24], that our map induces* * a homotopy equivalence from the homotopy fiber of the map TkF ! Tk-1F to the homotopy fibe* *r of the map TksF ! Tks-1F , for all k. Our assertion follows by induction on k. Recall the category fOs(W ) defined in Section 7. Let fOs(M) be the full subc* *ategory of fOs(W ) consisting of objects (OE, m) such that OE(m) is a subset of M. Define also fOs* *k(M) to be the full subcategory of fOs(W ) consisting of objects (OE, m) such that m is at most k. Recall the functor ss :fOs(W ) ! Os(W ), (OE, m) 7! OE(m), defined in Section* * 7. It is clear that this functor restricts to a functor ss :fOsk(M) ! Osk(M). Recall also the * *notion of a right cofinal functor between small categories, as defined by Bousfield and Kan in [4* *, Chapter XI, x9]. The importance of this notion for us is that right cofinal functors preserve ho* *motopy limits of contravariant functors ([4, Theorem XI.9.2]). Lemma 9.2. The functor ss :fOsk(M) ! Osk(M) is right cofinal. Proof.Given an object U 2 Osk(M), we need to prove the contractibility of the u* *nder-category U # ss, which is exacly the full subcategory of fOsk(M) consisting of objects (* *OE, m) such that U OE(m). This subcategory is contractible because it has a (non-unique) initi* *al object, namely any object (OE, mU) such that OE(mU) = U where mU is the number of connected co* *mponents of U (there are mU! such objects). We can now prove the main result of this section. Recall from Section 7 the f* *unctor B(~(-), V ): fOs(W ) ! Top which by abuse of notation we denote by (OE, m) 7! B(m, V ). Theorem 9.3. Let W V be an inclusion of Euclidean vector spaces, let M be an * *open sub- manifold of W , and let k be a natural number. If dimV > 2 dimW then there is a* *n equivalence of spectra ____ 1Y ____ 1Y ____ TkHQ ^ Emb (M, V )+ ' Tk|| HiEmb (M, V )|| ' holim || Hi(Emb (ss(OE* *, m), V )))|| i=0 i=0(OE,m)2fOsk(M) where || Hi(X)|| is the Eilenberg-Mac Lane spectrum that has the i-th rational * *homology of X in degree i. Proof.By Proposition 9.1 and Lemma 9.2 we have ____ ____ TkHQ ^ Emb (M, V )+ ' holim HQ ^ Emb (ss(OE, m), V ))+. (OE,m)2fOsk(M) ____ By Proposition 7.1, the functors Emb (ss(OE, m), V ) and B(~(OE, m), V ) = B(m,* * V ) are weakly equiv- alent, as functors on fOsk(W ). It follows that their restrictions to fOsk(M) a* *re weakly equivalent, and so ____ TkHQ ^ Emb (M, V )+ ' holim HQ ^ B(m, V )+. (OE,m)2fOsk(M) CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 25 Using the Quillen equivalence between rational spectra and rational chain compl* *exes, and the fact that homotopy limits are preserved by Quillen equivalences, we conclude th* *at there is a weak equivalence (or more precisely a zig-zag of weak equivalences) in ChQ ____ TkC *(Emb (M, V )) ' holim C*(B (m, V )). (OE,m)2fOsk(M) On the other hand, by Proposition 6.5 and Lemma 3.6, the functor m 7! C*(B (m, * *V )) R from Ofsk(M) to ChR is formal. By Remark 3.5 we get that ____ 1Y holim C *(Emb (ss(OE, m), V )) R ' holim H i(B (m, V ); R* *). (OE,m)2fOsk(M) i=0(OE,m)2fOsk(M) Recall that B(m, V ) is equivalent to the space of configurations of m points i* *n V and it only has homology in dimensions at most (m - 1)(dim(V ) - 1). Since m k, the prod* *uct on the right hand side of the above formula is in fact finite (more precisely, it is n* *on-zero only for i = 0, dim(V )-1, 2(dim(V )-1), . .,.(k-1)(dim(V )-1)). Therefore, we may think* * of the product as a direct sum, and so tensoring with R commutes with product in the displayed* * formulas below. By Proposition 8.3, we know that tensoring with R commutes, in our case, with h* *olim, and so we obtain the weak equivalence _ ! ____ Y1 TkC *(Emb (M, V )) R ' holim Hi(B (m, V ); Q) R i=0(OE,m)2fOsk(M) * * ____ It is well-known (and is easy to prove using calculus of functors) that spaces * *such as Emb (M, V ) are homologically of finite type, therefore all chain complexes involved are ho* *mologically of finite type. Two rational chain complexes of homologically finite type that are quasi-* *isomorphic after tensoring with R are, necessarily, quasi-isomorphic over Q. Therefore, we have * *a weak equivalence in ChQ. ____ 1Y TkC *(Emb (M, V )) ' holim H i(B (m, V ); Q) i=0(OE,m)2fOsk(M) The desired result follows by using, once again, Proposition 7.1 and the equiva* *lence between ChQ and rational spectra. 10.Formality and the splitting of the orthogonal tower In this section we show that Theorem 9.3, which is about the splitting of a_c* *ertain_lim-Postnikov tower, can be reinterpreted as the splitting of the orthogonal_tower_of HQ ^ Em* *b (M, V )+. Thus in this section we mainly focus on the functoriality of HQ ^ Emb (M, V )+ in V * *and, accordingly, terms like "Taylor polynomials", "derivatives", etc. are always used in the con* *text of orthogonal calculus1 ___________ 1We are committing a slight abuse of notation here, because the definition of* * ____Emb(M, V ) depends on choosing ____ a fixed embedding M ,! W, and therefore Emb(M, V ) is only defined_for vector s* *paces containing W. One way around this problem would be to work with the functor V 7! Emb(M, W V ). To a* *void introducing ever messier notation, we chose to ignore this issue, as it does not affect our arguments in* * the slightest. 26 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C ____ As we have seen, embedding calculus tells us, roughly speaking, that 1 Emb (* *M, V )+ can be written as a homotopy inverse limit of spectra of the form 1 C(k, V )+ where C* *(k, V ) is the space of configurations of k points in V . A good place to start is therefore * *to understand the orthogonal Taylor tower of V 7! 1 C(k, V )+. The only thing that we will need * *in this section is the following simple fact (we will only use a rationalized version of it, but i* *t is true integrally). Proposition 10.1. The functor V 7! 1 C(k, V )+ is polynomial of degree k - 1.* * Assume dim(V ) > 1. For 0 i k - 1, the i-th layer in the orthogonal tower, Di 1 C* *(k, V )+, is equivalent to a wedge of spheres of dimension i(dim(V ) - 1). This proposition is an immediate consequence of Proposition 10.3 below, and i* *ts rational version is restated more precisely as Corollary 10.4. We now digress to do a detailed c* *alculation of the derivatives of 1 C(k, V )+. First, we need some definitions. Definition 10.2. Let S be a finite set. A partition of S is an equivalence re* *lation on S. Let P (S) be the poset of all partitions of S, ordered by refinement (the finer the* * bigger). We say that a partition is irreducible if each component of has at least 2 elements. The geometric realization of the poset P (S), |P (S)|, is a contractible simp* *licial complex with a boundary @|P (S)|. The boundary consists of those simplices that do not conta* *in the morphism from the initial object of P (S) to the final object as a 1-dimensional face. L* *et TS be the quotient space |P (S)|=@|P (S)|. There is a well-known equivalence [18, 4.109], ` TS ' S|S|-1. (|S|-1)! If S = {1, . .,.n}, we denote P (S) by P (n) and TS by Tn. Now let be a partition of S = {1, . .,.n}, and let P ( ) be the poset of al* *l refinements of . Define T as before, to be the quotient |P ( )|=@|P ( )|. It is not hard to * *see that if is a partition with components (~1, . .,.~j) then there is an isomorphism of posets P ( ) ~=P (~1) x . .x.P (~j) and therefore a homeomorphism T ~=T~1^ . .^.T~j. In particular, T is equivalent to a wedge of spheres of dimension n - j. We ca* *ll this number the excess of and denote it by e( ). Proposition 10.3. For i > 0, the i-th layer of 1 C(k, V )+ is equivalent to ` Di 1 C(k, V )+ ' Map* T , 1 SiV { 2P(k)|e( )=i} where the wedge sum is over the set of partitions of k of excess i. Proof.Denote the fat diagonal of kV by kV := {(v1, . .,.vk) 2 kV :vi = vj for * *somei 6= j}. The smashed-fat-diagonal of SkV is kSV := {x1^ . .^.xk 2 ^ki=1SV = SkV : xi= xj for somei 6= j}. Thus C(k, V ) = kV \ kV = ((kV ) [ {1}) \ (( kV ) [ {1}) = SkV \ kSV . CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 27 Recall that for a subpolyhedron in a sphere, j :K ,! Sn, Spanier-Whitehead du* *ality gives a weak equivalence of spectra 1 (Sn \ K)+ ' Map *(Sn=K, 1 Sn) which is natural with respect to inclusions L K and commutes with suspensions* *. In our case Spanier-Whitehead duality gives an equivalence 1 C(k, V )+ ' Map *(SkV= kSV , 1 SkV) which is natural with respect to linear isometric injections. The right hand si* *de is equivalent to the homotopy fiber of the map Map *(SkV, 1 SkV) -! Map *( kSV , 1 SkV) Since Map *(SkV, 1 SkV) ' 1 S0 is a constant functor, it has no layers of deg* *ree greater than zero. Therefore, for i > 0, i j Di 1 C(k, V )+ ' DiMap *( kSV , 1 SkV) It is not hard to see (see [2], Lemma 2.2 for a proof) that kSV can be "filter* *ed" by excess. More precisely, there is a sequence of spaces k1SV -! k2SV -! . .-.! kk-1SV = kSV such that the homotopy cofiber of the map ki-1SV ! kiSV is equivalent to ` K ^ S(k-i)V { 2P(k)|e( )=i} where K is a de-suspension of T . It follows that Map *( kSV , 1 SkV) can be * *decomposed into a finite tower of fibrations Map *( kSV , 1 SkV) = Xk-1 -! Xk-2 -! . .-.! X1 where the homotopy fiber of the map Xi! Xi-1is equivalent to Map *(K , 1 SiV) Since this is obviously a homogeneous functor of degree i, it follows that Xi i* *s the i-th Taylor polynomial of Map *( kSV , 1 SkV). The proposition now follows easily. Rationalizing, we obtain the following corollary. Corollary 10.4. Each layer in the orthogonal tower of the functor V 7! HQ ^ C(k* *, V )+ is an Eilenberg-Maclane spectrum. More precisely, ( || Hi(dim(V )-1)(C (k,iVf))||i k - 1; Di(HQ ^ C(k, V )+) ' *, otherwise where || Hi(dim(V )-1)(C (k, V ))|| is the Eilenberg-Mac Lane spectrum that has* * the i(dim(V ) - 1)-th rational homology of C(k, V ) in degree i(dim(V ) - 1). Therefore, this orthogonal tower coincides, up to indexing, with the Postniko* *v tower, i.e. Pn(HQ ^ C(k, V )+) ' Pod(n)(HQ ^ C(k, V )+), where d(n) is any number satisfying n(dim V - 1) d(n) < (n + 1)(dim V - 1). 28 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C Proof.The computation of the layers is an immediate application of the previous* * proposition. Set X = HQ ^ C(k, V )+ and consider the following commutative square X _______//_Pod(X) | | | | |fflffl fflffl| Pn(X) ____//_PodPn(X). A study of the homotopy groups of the layers shows that the bottom and the righ* *t maps are weak equivalences when d is the prescribed range. We will also need the following proposition. Proposition 10.5. For every n there exists a large enough k such that the natur* *al map ____ ' ____ PnHQ ^ Emb (M, V )+ -! holimPnHQ ^ Emb (U, V )+ U2Ok(M) is an equivalence. The same holds if Pn is replaced by Dn. ____ Proof.We will only prove the Pn version. The target_of_the map is TkPnHQ ^ Emb* * (M, V )+. Applying Lemma 8.6 to the functor E(M,_V ) = HQ ^ Emb (M,_V_)+, it is enough to* * prove that for a large enough k the map PnHQ^Emb (M, V )+ ! PnTkHQ^Emb (M, V )+ is an equi* *valence. Consider again the formula for the k-th layer in the embedding tower ` ` ' ' M c , HQ ^"C(k, V )+. k It is not hard to see that the spectrum_HQ_^"C(k, V )+is roughly_k_dim(V_)2-con* *nected_(exercise for the reader). It follows that HQ ^ Emb (M, V )+ and TkHQ ^ Emb (M, V )+ agr* *ee to order roughly_k_2(in the sense_defined_in Section 8). It follows, by Proposition 8.5* *, that the map HQ^Emb (M, V )+ ! TkHQ^Emb (M, V )+ induces an equivalence on Pn, for roughly n* * k_2. We are now ready to state and prove our main theorem ____ Theorem 10.6. Suppose dimV > 2 dimW . Then the orthogonal tower for HQ ^ Emb (M* *, V )+ splits. In other words, there is an equivalence ____ Yn ____ PnHQ ^ Emb (M, V )+ ' DiHQ ^ Emb (M, V )+. i=0 ____ Proof.By Lemma 8.6 and Proposition 10.5, and using the model for TkHQ ^ Emb(M, * *V )+ given in Theorem 9.3, it is enough to show that _ ! n Y Pn holim HQ ^ B(m, V )+ ' holim Di(HQ ^ B(m, V )+). (m,OE)2fOsk(M) i=0(m,OE)2fOsk(M) By Corollary 10.4 the Taylor tower of HQ ^ B(m, V )+ coincides, up to regrading* *, with the Postnikov tower. By the proof of Theorem 9.3, the homotopy limit holimHQ ^ B(m,* * V )+ splits as a product of the homotopy limits of the layers in the Postnikov towers. Sin* *ce diagrams of layers in the Postnikov towers and diagrams of layers in the orthogonal towers * *are diagrams of Eilenberg-MacLane spectra that are equivalent on homotopy groups, they are equi* *valent diagrams CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 29 (as per Remark 3.4). It follows that holimHQ ^ B(n, V )+ splits as a product of* * the homotopy limits of the layers in the orthogonal towers. It is easy to see that the splitting is natural with respect to embeddings of* * M, but notice that we do not claim that the splitting is natural in V . 11. The layers of the orthogonal tower In this section_we explicitly describe the layers (in the sense of orthogonal* * calculus) of the Taylor tower of HQ ^ Emb (M, V ) as the twisted cohomology of certain spaces of partit* *ions attached to M. We will try to give a "plausibility argument" for our formulas, but a detai* *led proof will appear in [1]. We encountered partition posets in Section 10. Here, however, we need to cons* *ider a different category of partitions. If is a partition of S, we call S the support of . W* *hen we need to emphasize that S is the support of , we use the notation S( ). Also, we denote* * by c( ) the set of components of . Then can be represented by a surjection S( )!!c( ). L* *et C be the mapping cylinder of this surjection. Then S( ) C . In the previous sectio* *n we defined the excess of to be e( ) := |S( )| - |c( )|. It is easy to see that e( ) = rank(H 1(C , S( )). Let 1, 2 be partitions of S1, S2 respectively. A "pre-morphism" ff : 1 ! * *2 is defined to be a surjection (which we denote with the same letter) ff : S1!!S2 such that 2 is* * the equivalence relation generated by ff( 1). It is easy to see that such a morphisms induces * *a map of pairs (C 1, S( 1)) ! (C 2, S( 2)), and therefore a homomorphism ff* : H1(C 1, S( 1)) -! H1(C 2, S( 2)). We say that ff is a morphism if ff* is an isomorphism. In particular, there can* * only be a morphism between partitions of equal excess. Roughly speaking, morphisms are allowed to * *fuse components together, but are not allowed to bring together two elements in the same compon* *ent. For k 2, let Ek be the category of irreducible partitions (recall that is* * irreducible if none of the components of is a singleton) of excess k, with morphisms as defined a* *bove. Notice that if is irreducible of excess k then the size of the support of must be betwe* *en k + 1 and 2k. Next we define two functors on Ek - one covariant and one contravariant. Rec* *all from the previous section that P ( ) is the poset of refinements of . A morphism ! 0* *induces a map of posets P ( ) ! P ( 0). It is not difficult to see that this map takes bounda* *ry into boundary, and therefore it induces a map T ! T 0. This construction gives rise to a func* *tor Ek ! Top, given on objects by 7-! T . In fact, to conform with the classification of homogeneous functors in orthog* *onal calculus, we would like to induce up T to make a space with an action of the orthogonal gro* *up O(k). Let eT := Iso(Rk, H1(T ( ), S( ); R))+ ^ T where Iso(V, W ) is the space of linear isometric isomorphisms from V to W (thu* *s Iso(V, W ) is abstractly homeomorphic to the orthogonal group if V and W are isomorphic, and* * is empty otherwise). In this way we get a functor from Ek to spaces with an action of O(* *k). 30 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C The other functor (a contravariant one) that we need is 7-! MS( )= (M) where (M) is the space of maps from S( ) to M that are non-injective on at le* *ast one component of . If is the partition with one component then (M) is the usua* *l fat diagonal. It is not hard to see that this defines a contravariant functor from Ek to spac* *es. Let M[ ]:= MS( )= (M). Consider the "tensor product" (homotopy coend) eT EkM[ ], which is a space with an action of O(k). ____ Theorem 11.1. The k-th layer of the orthogonal calculus tower of 1 Emb (M, V )* *+ is equivalent to i jO(k) Map * eT EkM[ ], 1 SV k . Idea of proof.Embedding calculus suggests that it is almost enough to prove_the* *_theorem in the case of M homeomorphic to a finite disjoint union of balls. In this case Emb (M* *, V ) is equivalent to the configuration space C(k, V ). It is not hard to show that then the formula * *in the statement of the current theorem is equivalent to the formula given by Proposition 10.3. The* * current theorem just restates the formula of Proposition 10.3 in a way that is well-defined for* * all M. ____ It follows that the k-th layer of HQ ^ Emb (M, V )+ is given by the same form* *ula as in the theorem, with 1 replaced with HQ^. Corollary 11.2. Suppose that f : M1 ! M2 is a map inducing an isomorphism in ho* *mology. Then for each n, the n-th layers of the orthogonal towers of the two functors ____ V 7! 1 Emb (Mi, V )+, i = 1, 2 are homotopy equivalent. Similarly, if f induces_an isomorphism in rational hom* *ology then the layers of the orthogonal towers of V 7! HQ ^ Emb (Mi, V )+ are equivalent. Proof.It is not hard to show that eT EkM[ ]is a finite CW complex with a free a* *ction (in the pointed sense) of O(k). Since the action is free, the fixed points construction* * in the formula for the layers in the orthogonal tower can be replaced with_the homotopy fixed poin* *ts construction. Thus, the k-th layer in the orthogonal tower of 1 Emb (M, V )+ is equivalent to i jhO (k) Map * Te EkM[ ], 1 SV k . It is easy to see that this is a functor that takes homology equivalences in M * *to homotopy equivalences. For the rational case, notice that i j hO (k) Map* eT EkM[ ], HQ ^ SV k is a functor of M that takes rational homology equivalences to homotopy equival* *ences. Some remarks are perhaps in order. CALCULUS, FORMALITY, AND EMBEDDING SPACES * * 31 Remark 11.3. There is a description of T as a space of rooted trees (more prec* *isely, forests). For a detailed discussion of the relationship between spaces of partitions and spac* *es of rooted trees we refer the reader, once again, to [6]. We do not really need this here, but_such* *_a reformulation is very convenient if one wants to extend the results of this section to 1 Emb (M* *, N) for a general manifold N. There is an analogous description, which is to some extent similar_* *in spirit, but is both more complicated and more interesting, of the layers of the functor Emb (M* *, N x V ). The construction involves certain spaces of graphs (as opposed to just trees). All * *this will be discussed in more detail in [1]. Remark 11.4. It may be helpful to note that the space eT EkM[ ]can be filtered* * by the size of support of (that is, by the number of points_in_M involved). This leads to* * a decomposition of the k-th layer in the orthogonal tower of 1 Emb (M, V ) as a finite tower o* *f fibrations, with k terms, indexed k + 1 i 2k, corresponding to the number of points in M. T* *his is the embedding tower of the k-th layer of the orthogonal tower. For example, the sec* *ond layer of the orthogonal tower fits into the following diagram, where 2,2M is the singular s* *et of the action of 2o 2 on M4, the left row is a fibration sequence, and the square is a homotop* *y pullback. 4 1 2V o Map *(_M__ 4M^ T2^ T2, S ) 2 2 | | __fflffl|_ 4 D2 1 Emb (M, V )___________//Map*(__M__ 2,2M^ T2^2, 1 S2V) 2o* * 2 | | | | fflffl| fflffl| 3 1 2V M3 ^2 1 2V Map *(_M__ 3M^ T3, S__)_3____//Map*(____ 3M^ T2 , S ) 2 Remark 11.5. To relate this to something "classical", note that the top layer o* *f the embedding tower of the k-th layer of the orthogonal tower is Map*(M2k= 2kM ^ T2^k, 1 SkV) 2o k. This is the space of "chord diagrams" on M, familiar from knot theory. In fact,* * in the case of M being a circle (or an interval, in which case one considers embeddings fixed ne* *ar the boundary), it is known from [17] that the Vassiliev homology spectral sequence, which also co* *nverges_to the space of knots, collapses at E1. Thus the orthogonal tower spectral sequence for HQ^E* *mb (M, V ) must coincide with Vassiliev's. It is not hard to verify directly that the two E1 te* *rms are isomorphic (up to regrading). ____ Finally, we deduce the rational homology invariance of Emb (M, V ). Theorem 11.6. Let M and M0 be two manifolds such that there is a zig-zag of map* *s, each inducing an isomorphism in rational homology, connecting M and M0. If dim V 2 max(ED (M), ED(M0)), ____ ____ then Emb (M, V ) and Emb (M0, V ) have the same rational homology groups. 32 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C References [1]G. Arone. The derivatives of embedding functors. In preparation. [2]G. Arone. 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Weiss. Embeddings from the point of view of immersion theory I. Geom. To* *pol., 3: 67-101, 1999. [25]M. Weiss. Homology of spaces of smooth embeddings. Quart. J. Math., 55: 499* *-504, 2004. Department of Mathematics, University of Virginia, Charlottesville, VA E-mail address: zga2m@virginia.edu Institut Math'ematique, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgi* *um E-mail address: lambrechts@math.ucl.ac.be Department of Mathematics, University of Virginia, Charlottesville, VA E-mail address: ismar@virginia.edu