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
bifunctor (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 noncompact, 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 GoodwillieWeiss 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 (cofunctors)
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 () ! Tk1F () ! . ..! 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 (GoodwillieKlein,_[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 kUtuples of distinct points*
* in V or, equivalently,
the space of kUtuples 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 welldefined, 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 quasiisomorphisms 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 nth Taylor polynomial of G in the orthogonal sense. Let DnG(V*
* ) denote the
nth homogeneous layer in the orthogonal Taylor tower, namely the fiber of the *
*map PnG(V ) !
Pn1G(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 Mvariable (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 nth 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 nth 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 zigza*
*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 realvalued 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 codimension 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 highdimensional 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 wellknown that there is a rational equivalence HQ *
*^ X+ ' o1pX+.
o V will denote the category of rational vector spaces (or Qvector spaces),*
* and V the
category of simplicial Qvector spaces.
o ChQ and Ch Rwill denote the category of nonnegatively 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 quasiisomorphisms, 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 ringspectrum) 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 nth 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 Qvector
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 ndimensional
boundaries in C. We define the nthPostnikov 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 degreewise surjection) ssn*
*: Pon(C) i Pon1(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 = aen1. 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 Adiagram *
*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 Adiagrams 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 fibrantcofibrant 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 zigzag
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 Adiagram 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 Adiagram of chain c*
*omplexes,
F :A ! Ch, splits homogeneously if there exist Adiagrams {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 Adiagram of chain complexes is *
*formal if and
only if it splits homogeneously.
CALCULUS, FORMALITY, AND EMBEDDING SPACES *
* 9
Proof.Let F be an Adiagram 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 Pon1(Fn)
is concentrated in degrees n and n + 1 and its homology is exactly Hn(F ). Thus*
* we have a chain
of quasiisomorphisms
Fn '!Po n(Fn) ' ker(Pon(Fn) ! Pon1(Fn))'!H n(ker(Pon(Fn) ! Pon1(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 Adiagrams 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
EilenbergMac Lane spectra: If F and G are two Adiagrams of EilenbergMac 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 nth stage is holimPonF . We call it the limPostnikov 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 limPostnikov 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*
*0diagram
of chain complexes. If the A0diagram F is formal then so is the Adiagram ~*(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 nonnegatively 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)pfn1dY },
for fn 2 hom(Yn, Zn+p).
4.2. Enriched categories. A category O enriched over C, or a Ccategory, consis*
*ts of a class I
(representing the objects of O), and, for any objects i, j, k 2 I, a Cobject O*
*(i, j) (representing
the morphisms from i to j in O) and Cmorphisms
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 CIcategory to signify a category enriched over C, whose *
*set of objects is I.
Let O be a CIcategory and R be a category enriched over C. A (covariant) fun*
*ctor enriched
over C, or Cfunctor from O to R,
M :O ! R,
consists of an Robject 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
Cfunctor.
A natural transformation enriched over C, : M ! M0, between two Cfunctors M*
*, M0:O !
R consists of Cmorphisms
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 CIcategories as a category *
*in its own right. A
morphism of CIcategories 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 CIcategories 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 CIcatego*
*ry O,
a (right) Omodule is a contravariant Cfunctor from O to C. Explicitly an Omo*
*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 Cmorphisms
M(j) O(i, j) ! M(i)
which are associative and unital. A morphism of Omodules, : M ! M0, is an e*
*nriched
natural transformation, i.e., a collection of Cmorphism (i): M(i) ! M0(i) sat*
*isfying the usual
naturality requirements. Such a morphism of Omodule is a weak equivalence if *
*each (i) is
a weak equivalence in C. We denote by Mod O the category of right Omodules an*
*d natural
transformations.
Let : O ! R be a morphism of CIcategories. 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 CIcategories, 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 ChIcategories
to denote categories enriched over chain complexes, with object set I. Note tha*
*t the category of
modules over a ChIcategories admits coproducts (i.e. direct sums).
Theorem 4.1 (SchwedeShipley, [20]).
(1) Let O be a ChIcategory. 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 ChIcategories. 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 CIcategories 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 CIcategories : *
*O ! R and a
morphism of Omodules : M ! *(N). The corresponding category of pairs (O, M) *
*is called
the CImodule category. A morphism ( , ) in CImodule is called a weak equival*
*ence if both
and are weak equivalences. Two objects of CImodule 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
Omodules 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 ChImodules. If M*
* is weakly
equivalent as an Omodule to a direct sum Mn, then M0 is weakly equivalent as *
*an O0module
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 Rmodules. It is clear that the r*
*estriction of
scalars functor * preserves direct sums and weak equivalences (quasiisomorphi*
*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 Omodule 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
Omodule 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 CIcategories to DIcategories. Explicitly if O is a CIcategory then F (O*
*) is the Dcategory
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 TopIcategory then C*(B) and H*(B) are ChIcategories. A*
*lso if B :B !
Top is a Bmodule then C*(B) is a C*(B)module and H*(B) is an H*(B)module. We*
* also have
the ChIcategory 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 Adiagram 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 Topenriched 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 Cfunctor between two Ccategories. 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 Omodule M induces an Offid*
*iagram Mffiin C
and a morphism of Omodules induces a morphism of Offidiagrams.
Let F :C ! D be a lax symmetric monoidal functor, let O be a CIcategory, and*
* let M :O ! C
be an Omodule. As explained before, we have an induced DIcategory 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 CIcate*
*gory and let
M be an Omodule. 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 CIcategory. If :*
* M '! M0
is a weak equivalence of Omodules then ffi:Mffi'!M0ffiis a weak equivalence *
*of Offidiagrams.
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 zeroth term,
while we consider operads with one. Briefly, an operad in a symmetric monoidal *
*category (C, , 1),
or a Coperad, 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*
* Coperads 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 Coperad 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 CNcategory is *
*a category enriched
over C whose set of objects is N. The CNcategory associated to the Coperad O(*
*o) is the category
O defined by a
O(m, n) = O(ff1(1)) . . .O(ff1(n))
ff:m_!n_
where the coproduct is taken over set maps ff: m_ := {1, . .,.m} ! n_ := {1, . *
*.,.n} and
O(ff1(j)) = O(mj) where mj is the cardinality of ff1(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 Coperad and let O be the associated CNcategory. A right modul*
*e (in the sense
of operads) M(o) over O(o) gives rise to a right Omodule (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 Cmorphisms
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(ff1(1)) . . .O(ff1(n)) ! M(m).
ff:m_!n_
If f :O(o) ! R(o) is a morphism of operads then we have an associated right Om*
*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 CNcategories 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 CNmodules, 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 CNcategory associated to O(o). Then F (O(o)) is an operad in D, a*
*nd F (O) is a
DNcategory. It is easy to see that there is a natural morphism from the DNcat*
*egory associated
to the Doperad 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 Omodule 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 wellknown oper*
*ad in (Top, x, *),
consisting of the topological spaces
B (n, V ) = {ntuples 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 TopNcategory associated to the standard balls operad B(o, V ) will be de*
*noted by B(V ).
An object of B(V ) is a nonnegative 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 ChNcategories 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 FultonMacPherson 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 quasiisomorphism
: SemiAlgChain*(FM d(n)) '!Graphsd(n) ^R
where SemiAlgChain*is a chain complex of semialgebraic chains naturally quasi*
*isomorphic to
singular chains and Graphsdis the chain complex of admissible graphs defined in*
* [14, Definition
13]. For , a semialgebraic 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 nonzero semial*
*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 Icategory. We say that an Omodule M :O ! Ch s*
*plits
homogeneously if there exists a sequence {Mn}n2N of Omodules 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 ChNcategory and we have an obvious inclu*
*sion functor
(because our chain complexes are nonnegatively 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 Ncategories, 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 abovementioned 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 ChNmodule 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 )ffidiagram B(, V )f*
*fi.
Proposition 6.5. If dimV > 2 dimW then the B(W )ffidiagram
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 )ffidiagram 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 )))ffidiagr*
*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 ))ffidiagram 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
mtuple 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 bifunctors 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
cocartesian if each of its twodimensional faces is a homotopy pushout square.*
* A cofunctor F on
O(M) is called polynomial of degree k if it takes strongly cocartesian k + 1d*
*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 kth Taylor polynomial of F . *
*F is said to be
homogeneous of degree k if it is polynomial of degree k and Tk1F is equivalent*
* to the trivial
functor. For each k, there is a natural map TkF ! Tk1F , compatible with the m*
*aps F ! TkF
and F ! Tk1F . Its homotopy fiber is a homogeneous functor of degree k, and it*
* is called the
kth layer of the tower. It plays the role of the kth term in the Taylor serie*
*s of a function. For
spacevalued functors, there is a useful general formula for the kth 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 ktupl*
*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 kdimensional 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 ! Tk1F 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 nondegenerate 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 zigzag 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*
*gzag 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 ! Pn1G and its fiber (again called the *
*nth layer) is
denoted by DnG. DnG is a homogeneous functor, in the sense that it is polynomia*
*l of degree n
and Pn1DnG ' *. 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 *
*onepoint 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 (spectrumvalued) 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 nth derivative of G. There is a useful description of the derivativ*
*es of G as stabilizations
of certain types of iterated crosseffects 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 nth order via ff if the map ff(V ) : G1(V ) ! G2(V ) is (n + 1) di*
*m(V ) + cconnected,
where c is a possibly negative constant, independent of V . Using the descripti*
*on of derivatives in
terms of crosseffects, it is easy to prove the following proposition
Proposition 8.5. Suppose that G1 and G2 agree to nth 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 nth 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 nth Taylor polynomial in the orthogonal sense. Similarly, TkE(M, V *
*) is the functor
obtained by taking the kth 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 kth 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) ! Tks1F (M)
24 GREGORY ARONE, PASCAL LAMBRECHTS, AND ISMAR VOLI'C
induced by the inclusion of categories Osk1(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 ! Tk1F to the homotopy fibe*
*r of the map
TksF ! Tks1F , 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*
*ndercategory
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 (nonunique) 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 EilenbergMac Lane spectrum that has the ith 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 zigzag 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*
*onzero only for
i = 0, dim(V )1, 2(dim(V )1), . .,.(k1)(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 wellknown (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, quasiisomorphic 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_limPostnikov
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 ith 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 1dimensional face. L*
*et TS be the quotient
space P (S)=@P (S). There is a wellknown equivalence [18, 4.109],
`
TS ' SS1.
(S1)!
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 ith 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 smashedfatdiagonal 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, SpanierWhitehead 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
SpanierWhitehead 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 ! . ..! kk1SV = kSV
such that the homotopy cofiber of the map ki1SV ! kiSV is equivalent to
`
K ^ S(ki)V
{ 2P(k)e( )=i}
where K is a desuspension of T . It follows that Map *( kSV , 1 SkV) can be *
*decomposed into
a finite tower of fibrations
Map *( kSV , 1 SkV) = Xk1 ! Xk2 ! . ..! X1
where the homotopy fiber of the map Xi! Xi1is equivalent to
Map *(K , 1 SiV)
Since this is obviously a homogeneous functor of degree i, it follows that Xi i*
*s the ith 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
EilenbergMaclane 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 EilenbergMac 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 kth 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_)2con*
*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
EilenbergMacLane 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 "premorphism" 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 noninjective 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 kth 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 welldefined for*
* all M.
____
It follows that the kth 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 nth 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 kth 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 kth 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 kth 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 kth 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 zigzag 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
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Department of Mathematics, University of Virginia, Charlottesville, VA
Email address: zga2m@virginia.edu
Institut Math'ematique, 2 Chemin du Cyclotron, B1348 LouvainlaNeuve, Belgi*
*um
Email address: lambrechts@math.ucl.ac.be
Department of Mathematics, University of Virginia, Charlottesville, VA
Email address: ismar@virginia.edu