OPERADS AND KNOT SPACES
DEV P. SINHA
1.Introduction
Let Em denote the space of embeddings of the interval I = [-1, 1] in the cube*
* Im with endpoints
and tangent vectors at those endpoints fixed on opposite faces of the cube, equ*
*ipped with an isotopy
through immersions to the unknot - see Definition 5.1. By Proposition 5.17, Em *
*is homotopy equivalent
to Emb (I, Im ) x Imm(I, Im ). In [26], McClure and Smith define a cosimplicia*
*l object Oo associated to
any operad with multiplication O, whose homotopy invariant totalization we deno*
*te gTot(Oo) - see Defini-
tion 2.16 and Definition 2.5 below. Let Km denote the mth Kontsevich operad, in*
*troduced in [21], whose
entries are compactified configuration spaces and which is weakly equivalent to*
* the little k-cubes operad
[35] - see Definition 4.1 and Theorem 4.5 below.
Theorem 1.1. The totalization of the Kontsevich operad gTot(Kom) is homotopy eq*
*uivalent to theninverse
limit of the Taylor tower approximations for Em in the calculus of embeddings. *
*Moreover, gTot(Kom) is the
nth degree approximation.
Building on work of Goodwillie, Klein and Weiss [44, 15, 19, 18], and Volic [*
*41, 42], we have the following.
Corollary 1.2. For m > 3, Em is weakly equivalent to gTot(Kom). For m = 3, all*
* rational finite-type
invariants of framed knots factor through a map from Em to gTot(Kom).
Applying the homology spectral sequence of a cosimplicial space, we have the *
*following.
Corollary 1.3. For m > 3, there is a spectral sequence with E2 page given by th*
*e Hochschild cohomology
of the degree m - 1 Poisson operad which converges to the homology of Em .
These results resolve conjectures of Kontsevich from his address at the AMS M*
*athematical Challenges
Conference in the summer of 2000 [22]. Kontsevich's insights had already motiva*
*ted Tourtchine to give an
algebraic description of the E2-term of Vassiliev's spectral sequence along the*
* lines of Corollary 1.3 in [38].
Our results are at the level of spaces and show that the disagreement which Tou*
*rtchine found between
Hochschild cohomology of the Poisson operad and Vassiliev's E2-term is accounte*
*d for by the difference
between Em and the classical knot space.
Our results bring together some recent developments in algebraic topology and*
* its application to fields
such as deformation theory and knot theory. In [34] we presented models for spa*
*ces of knots, including a
cosimplical model which is analogous to the cosimplicial model for loop spaces.*
* We build on these results
in proving Theorem 1.1. In [26] McClure and Smith resolved the integral Delign*
*e conjecture, showing
that the totalization of an operad with multiplication has a two-cubes action b*
*oth in the setting of chain
complexes and that of spaces. We apply their results to establish the following.
Theorem 1.4. For any m, there is a little two-cubes action on gTot(Kom). For m *
*> 3, Em is a two-fold
loop space.
___________
1991 Mathematics Subject Classification. 57Q45, 18D50, 57M27.
Key words and phrases. knot spaces, operads, embedding calculus.
The author is supported by NSF grant DMS-0405922.
1
2 DEV P. SINHA
We conjecture that this two-cubes action on gTot(Kom) is compatible with a tw*
*o-cubes action on the space
of framed knots which has been recently defined by Budney [7], who goes on two *
*show that long knots
in dimension three are free over the two-cubes action. In future work we plan t*
*o investigate analogues
of this freeness result in higher dimensions. A first step will be to construct*
* operations compatible with
this the two-cubes structure in the homology spectral sequence for an operad wi*
*th multiplication, as
McClure and Smith currently plan to do. On the E1-term such operations will pre*
*sumably coincide with
Tourtchine's bracket, defined combinatorially in [38], but through their space-*
*level construction would also
be compatible with differentials and extend to further terms.
Some of the technical results developed in this paper may be of independent i*
*nterest. We fully develop
the operad structure on the simplicial compactification of configurations in Eu*
*clidean spaces. An operad
structure on the canonical (Axelrod-Singer) compactification is known [13, 23],*
* but does not yield an operad
with multiplication and instead admits a map from Stasheff's A1 operad. Our app*
*roach to this operad
structure blends geometry and combinatorics, revealing an operad structure on t*
*he standard simplicial
model for the two-sphere.
1.1. Acknowledgements. We thank J. McClure and J. Smith for their interest in t*
*his project, answers
to questions, and especially for writing Section 15 of [27]. We thank M. Markl *
*and J. Stasheff for comments
on early versions of this work, and M. Kontsevich for helpful conversations.
Contents
1. Introduction *
* 1
1.1. Acknowledgements *
* 2
2. Background material *
* 2
2.1. Cosimplicial spaces *
* 3
2.2. Operads *
* 5
3. The binomial operad *
* 7
4. The Kontsevich operad *
* 8
5. Models for spaces of knots and immersions arising from the calculus of em*
*beddings 12
5.1. A brief overview of the calculus of embeddings *
* 12
5.2. Knot space models through homotopy limits of configuration spaces *
* 13
5.3. A closer look at Em *
* 15
6. The main result *
* 16
7. Observations and consequences *
* 18
7.1. Spectral sequences *
* 18
7.2. A little two-cubes action from the McClure-Smith framework *
* 19
References *
* 20
2. Background material
Our main results are stated in terms of operads with multiplication and their*
* associated cosimplicial
spaces. As a chance to set the choice of definitions and notation which will be*
* most convenient, and as
an opportunity to place all standard material together, we review this material*
* here. For a more complete
survey we highly recommend [28]. A reader familiar with these constructions may*
* wish to skip this section
and refer back for clarification as needed.
OPERADS AND KNOT SPACES *
* 3
2.1. Cosimplicial spaces.
Definition 2.1. Let denote the category with one object for every non-negativ*
*e integer and where the
morphisms from k to ` are the order-preserving maps from [k] = {0, . .,.k} to [*
*`] = {0, . .,.`}, ordered in
the standard way. A cosimplicial object in a category C is a (covariant) functo*
*r from to C. A simplicial
object is a contravariant functor from to C.
Cosimplicial objects are denoted Xo, where Xk is the image of [k] under the f*
*unctor, also known as the
kth entry. Simplicial objects are denoted Xo. Central in the theory is the stan*
*dard cosimplicial space o,
whose kth entry is k, with vertices labelled by [k], and which sends a morphis*
*m [k] ! [`] to the linear
map which extends this map on vertices.
Every order-preserving map [k] ! [`] can be factored through elementary maps *
*di, which are an iso-
morphism but for one element - namely i - not in their image, and elementary ma*
*ps si, which are an
isomorphism but for having i and i + 1 in [k] both mapping to i 2 [`]. The corr*
*esponding maps between
entries of simplicial and cosimplicial objects, called (co)face and (co)degener*
*acy maps, are often taken as a
basis for their definition. The definitions are arranged so that the simplices *
*of a simplicial complex form a
simplicial set. Indeed, a simplicial set or simplicial space Xo may be realized*
*, denoted |Xo|, as the quotient
space of the union of Xix iover all i by the relations djx x fi ~ x x djfi and*
* sjx x fi ~ x x sjfi for all
x 2 Xi, fi 2 i.
Cosimplicial spaces naturally arise when studying mapping spaces. The totaliz*
*ation of a cosimplicial
space TotXo is the space of natural transformations from o to Xo, which is fir*
*st used tautologically to
study mapping spaces as follows.
Definition 2.2. For any X 2 C, a symmetric monoidal category with productJ , ta*
*king the product of
X with itself gives rise to a functor X- : Setop! C which sends S to s2SX. By*
* composing a simplicial
set Yo : ! Setopwith this functor, we obtain a cosimplicial object XYo.
Proposition 2.3. If Yo is a simplicial set and X is a space, Tot(XYo) is homeom*
*orphic to the space of
maps from |Yo| to X.
For based X and Yo we may replace XYn by the subspace of (x1, . .,.xff, . .).*
*, ff 2 Yn where if b denotes
the degenerate image of the basepoint of Yo in Yn we have that xb is the basepo*
*int of X. The resulting
cosimplicial space, which we denote XYo?, has totalization homeomorphic to the *
*space of based maps from
|Yo| to X. Another interesting example along these lines is the Hochschild cosi*
*mplicial vector space AS1o,
whose associated chain complex computes Hochschild cohomology of a ring A.
Cosimplicial spaces are intimately connected with homotopy limits (in fact, h*
*omotopy limits are defined
in terms of cosimplicial spaces in [5]). The nerve of a category C is the simpl*
*icial set Co, with Cibeing
the collection of i composable morphisms and structure maps defined through com*
*posing such maps or
inserting identity maps (see for example Chapter 14 of [20]). Denote the realiz*
*ation of the nerve of C by
BC, also called the classifying space. Recall that if c is an object of C, the *
*category C # c has objects
which are maps with target c and morphisms given by morphisms in C which commut*
*e with these structure
maps. The classifying space B(C # c) is contractible because C # c has a final *
*object, namely c mapping
to itself by the identity morphism. A morphism g from c to d induces a map from*
* C # c to C # d, so that
B(C # -) is itself a functor from C to spaces.
Definition 2.4. The homotopy limit of a functor E from a small category C to th*
*e category of spaces is
Nat(B(C # -), E), the space of natural transformations from B(C # -) to E.
For Xo fibrant in the standard model structure on cosimplicial spaces, that i*
*s those which satisfy the
matching condition 10.4.6 of [5], Theorem 11.4.4 of [5] states Tot(Xo) ' holimX*
*-o, an equivalence needed
for many applications. For cosimplicial spaces which do not necessarily satisfy*
* the matching condition, we
use an alternate definition of totalization for which this equivalence is a tau*
*tology.
4 DEV P. SINHA
Definition 2.5. oLet f obe the cosimplicial space whose [k]th entry is B( *
*# [k]) and whose
structure maps are the standard induced maps.
oFor a cosimplicial space Xo let gTotXo, called the homotopy invariant tot*
*alization, denote the space
of naturalktransformations from f oto Xo.
oLet gTotXo denote the space of natural transformations from the kth coske*
*leton of f oto Xo.
oLet k denote the full subcategory of whose objects are [i] for i k. *
*Let ik : k ! be the
inclusion functor.
In Section 15 of [27] the notation f oand gTotare used for any cofibrant repl*
*acement for o and the
corresponding totalization in the model structure on cosimplicial spaces where *
*all objects are fibrant (in
the usual model structure from [5], all objects are cofibrant). We choose parti*
*cular models for definiteness,
k
and so by definition gTot(Xo) is the homotopy limit of Xo and gTot(Xo) ~=holim(*
*-Xo O.ik)
The cosimplicial category is also intimately related to the category of sub*
*sets of a finite set.
Definition 2.6. Let P(k) be the category of all subsets of [k] = {0, . .,.k} wh*
*ere morphisms are defined
by inclusion. Let P0(k) be the full subcategory of non-empty subsets.
The connection of this category to the simplicial world is evident in the ide*
*ntification of BP0(k) with
the barycentric subdivision of a k-simplex. More specifically, we define maps e*
* k! k, and thus TotXo !
gTotXo for any Xo, as the induced map on classifying spaces for the functor whi*
*ch sends some [n] f![k] in
( # [k]) to the image of f, as a subset of [k]. There is also a translation be*
*tween cosimplicial diagrams
and those indexed by P0(k), which we will use in Section 6.
Definition 2.7. Let ck:P0(k) ! be the functor which sends a subset S to the o*
*bject in with the
same cardinality, and which sends and inclusion S S0to the composite [i] ~=S *
* S0~=[j], where [i] and
[j] are isomorphic to S and S0respectively as ordered sets.
*
* k
The following lemma is a consequence of Theorem 6.4 of [34], along with the o*
*bservation that gTot(Xo) =
holim(-ikO Xo).
*
* k
Lemma 2.8. For Xo a cosimplicial space, holim(-Xo O ck)is weakly equivalent to *
*gTotXo.
For any cosimplicial space there are spectral sequences for the homotopy grou*
*ps [5] and homology
groups [6, 33] of its totalization, which we will apply in Section 7. The homot*
*opy spectral sequence is
straightforward, with convergence immediate from its definition through the tow*
*er of fibrations
. . .TotiXo Toti+1Xo . .,.
whose homotopy inverse limit is TotXo. Unraveling the definitions we have the f*
*ollowing.
Proposition 2.9.TFor a cosimplicial space Xo there is a spectral sequence conve*
*rging to ß*(gTotXo)
with E1-p,q= kersk* ßq(Xp). The d1 differential is the restriction to this*
* kernel of the map
p+1i=0(-1)idi*:ßq(Xp-1) ! ßq(Xp).
The homology spectral sequence is more subtle, generalizing the Eilenberg-Moo*
*re spectral sequence.
One of the precise statements as to the convergence of this spectral sequence a*
*rising from [6] goes as
follows.
*
* T
Theorem 2.10. For a cosimplicial space Xo there is a spectral sequence with E1-*
*p,q= kersk*
Hq(Xp; Fp). The d1 differential is the restriction to this kernel of the map
p+1i=0(-1)idi*:Hq(Xp-1; Fp) ! Hq(Xp; Fp).
OPERADS AND KNOT SPACES *
* 5
This spectral sequence converges to H*(TotXo; Fp) if Xk is simply connected for*
* all k and E1-p,q= 0 when
q cp for some c > 1.
Alternately, one may arrive at the same spectral sequence from E2 forward wit*
*h E1-p,q= Hq(Xp) and
d1 defined as before, but not restricted to the kernel of the codegeneracies.
This theorem follows immediately from Theorem 3.2 of [6] as both of Bousfield*
*'s conditions, namely
that E1-p,q= 0 if p > q and that only finitely many E1-p,qwith q - p = n are no*
*n-zero for any given n,
follow from the vanishing with q cp for some c > 1.
These spectral sequences apply unchanged to the homotopy invariant totalizati*
*on. If Xo is a cosimpicial
space and Xo_is a fibrant replacement (as given by Proposition 8.1.3 and Theore*
*m 15.3.4 in [20]) then
i j i j
gTot(Xo) = Maps f o, Xo' Maps f o, Xo_' Maps( o, Xo_)= Tot(Xo_).
Because homotopy and homology of the entries and structure maps of Xo_agree wit*
*h those of Xo, the
identifications of the E1-terms of the associated spectral sequences are unchan*
*ged.
2.2. Operads. We choose to define non- operads in terms of a well-known [3, 24*
*, 34, 35] category of
rooted trees.
Definition 2.11. oA rooted, planar tree (or rp-tree) is an isotopy class of*
* finite connected acyclic
graph with a distinguished vertex called the root, embedded in the upper *
*half plane with the root
at the origin. Univalent vertices of an rp-tree (not counting the root, i*
*f it is univalent) are called
leaves.
oGiven an rp-tree T and a set of edges E the contraction of T by E is the *
*tree T0 obtained by, for
each edge e 2 E, identifying its initial vertex with its terminal vertex *
*(altering the embedding of
the tree in a neighborhood of e) and removing e from the set of edges.
oLet denote the category of rp-trees, where there is a unique morphism f*
*rom T to T0, denoted
either fT,T0or cE, if T0 is the contraction of T along some set of non-le*
*af edges E.
oEach edge of the tree is oriented by the direction of the root path, whic*
*h is the unique shortest
path to the root. The vertex of an edge which is further from the root is*
* called its inital vertex,
and the vertex closer to the root is called its terminal vertex. We say t*
*hat one vertex or edge lies
over another if the latter is in the root path of the former. A non-root *
*edge is called redundant if
its initial vertex is bivalent.
oBoth the collection of leaves in an rp-tree and the collections of edges *
*with a given terminal vertex
are ordered, using the clockwise orientation of the plane.
oA sub-tree of an rp-tree is a sub-graph consisting of a vertex v and some*
* collection of vertices lying
over v along with all edges for which these vertices are terminal, such t*
*hat the resulting subgraph
is connected. A sub-tree is an rp-tree through a linear isotopy which tra*
*nslates v to the origin.
See Figure 2.13 for some examples of objects and morphisms in . Let n denot*
*e the full subcategory
of trees with n leaves. Note that n differs from n of [34], which is naturall*
*y the full subcategory of
rp-trees without bivalent vertices. Each n has a terminal object, namely the u*
*nique tree with one vertex,
called the nth corolla fln as in [24]. We allow for the tree fl0 which has no l*
*eaves, only a root vertex, and is
the only element of 0. For a vertex v let |v| denote the number of edges for w*
*hich v is terminal, usually
called the arity of v.
Definition 2.12. A non- operad is a functor O from to a symmetric monoidal c*
*ategory (C, ) which
satisfies the following axioms.
(1)O(T) = v2TO(fl|v|).
(2)O(fl1) = 1C= O(fl0).
(3)If e is a redundant edge and v is its terminal vertex then O(c{e}) is the*
* identity map on v06=vF(flv0)
tensored with the isomorphism (1C -) under the decomposition of axiom (*
*1).
6 DEV P. SINHA
(4)If S is a subtree of T and if fS,S0and fT,T0contract the same set of*
* edges, then under the
decomposition of (1), F(fT,T0) = F(fS,S0) id.
We sketch the equivalence of this definition with the standard ones. By *
*axiom (1), the values of O are
determined by its values on the corollas O(fln), which corresponds to O(n)*
* in the usual operad terminology.
Axioms (2) and (3) correspond to the unit condition. By axiom (4), the val*
*ues of O on morphisms may
be computed by composing morphisms on sub-trees, so we may identify some s*
*ubset of basic morphisms
through which all morphisms factor. We illustrate some basic morphisms in *
* which correspond to the Oi
operations and May's operad structure maps in Figure 2.13. Another basic c*
*lass we consider is that of all
morphisms T ! fln where fln is a corolla and T is any tree; this class ext*
*ends May's structure maps.
Figure 2.13.
PSfrag_replacements_______
Two morphisms in which give rise to standard operad structu*
*re maps.
The first corresponds to a Oioperation, the second to one of May'*
*s structure maps.
Example 2.14. oThe associative operad A, defined in any symmetric mono*
*idal category, has
A(T) = 1C, and A(T ! T0) = id for all morphisms in .
oLet denote the full subcategory of rp-trees with no redundant edge*
*s (called the category of reduced
trees in [24]) and let P : ! denote the functor which contracts *
*all of the redundant edges of
an rp-tree. The operad of planar trees, Treenfrom Definition 1.41 of*
* [24], is the operad in the
category of sets which sends T to the set ( # P(T))of all T02 whi*
*ch map to P(T). It sends a
contraction of edges of T to the collection of contractions on the c*
*orresponding edges for trees over
P(T).
Definition 2.15. A map between operads is a natural transformation which r*
*espects the decomposition
of axiom (1) of Definition 2.12. An operad with multiplication is a non- *
*operad O equipped with a map
from the associative operad A.
The notion of operad with multiplication is due to Gerstenhaber and Voro*
*nov [12]. The canonical
example is the endomorphism operad of an associative algebra End(A). Algeb*
*ras over an operad with
multiplication are in particular associative algebras. An operad with mul*
*tiplication in the category of
spaces is an operad in the category of pointed spaces.
While we have taken a categorical approach to defining operads, we will *
*take a more coordinatized
approach to their associated cosimplicial objects. Recall the Oioperations*
* Oi: O(n) O(m) ! O(n+m-1)
which provide a basic set of morphisms for an operad, as illustrated in Fi*
*gure 2.13. From section 3 of [26]
we have the following.
Definition 2.16. oGiven an operad with multiplication O, let ~ denote *
*the morphism A(2) =
1C ! O(2).
oDefine di: O(n) ! O(n + 1) by
8
>><1C O(n) ~!idO(2) O(n) O0!O(n + 1) ifi = 0
di= >O(n) 1C id!~O(n) O(2) Oi!O(n + 1) if0 < i < n + 1
>: ~ id On+1
1C O(n) ! O(2) O(n) ! O(n + 1) ifi = n + 1.
OPERADS AND KNOT SPACES *
* 7
oDefine sias O(ci) where ci: fln ! fln-1 contracts the ith leaf of fln.
oLet Oo be the cosimplicial object in C whose nth entry is O(n) and whose *
*coface and codegeneracy
maps are given by di and si above. If C is the category of vector spaces *
*over a given field, let
HH*(O) be the homology of the cochain complex defined by the cosimplicial*
* abelian group Oo.
If C is the category of spaces, we call gTot(Oo) the totalization of Oo.
It is straightforward, and almost always left to the reader, to show that the*
* maps di and si satisfy
cosimplicial identities.
Remark. In the category of vector spaces, Tourtchine introduced the terminology*
* HH*(O) because if A
denotes an associative algebra and End(A) is its endomorphism operad then HH*(E*
*nd(A)) = HH*(A),
the usual Hochschild cohomology of A. We are not aware, however, of any sense *
*in which Hochschild
cohomology of operads is a cohomology theory for operads. Instead, Kontsevich c*
*onjectures that there
is a suitable enriched homotopy structure on the category of operads of spaces *
*such that gTot(Oo) is the
derived space of maps from the associative operad to O.
3.The binomial operad
*
* n
At the combinatorial heart of our work is the binomial operad, an intertwinin*
*g of the sets 2 , of
distinct pairs of elements (i, j) 2 n = {1, . .,.n} with i < j for definiteness*
*, with rooted planar trees as
follows. Recall that Setop, the opposite category to the category of pointed se*
*ts, is symmetric monoidal
with product is given by pointed union, denoted _, and unit given by the one-po*
*int set. Let S+ denote
the union of a set S with a disjoint base point.
Definition 3.1. oThe join of two leaves in an rp-tree is the first vertex (*
*that is, the farthest from
the root) at which their root paths coincide.
oLabel both the leaves of an rp-tree and the edges which emanate from a gi*
*ven vertex v with
elements of n and {1, . .,.|v|} respectively according to the order given*
* by planar embedding. To
an rp-tree T with n leaves and two distinct integers i, j 2 n let v be th*
*e join of the leaves labelled
i and j and define Jv(i), Jv(j) to be the labels of the edges of v over w*
*hich leaves i and j lie, as
illustrated below.
Figure 3.2.
oLet B, the binomialWoperad, be the non- operad in the category Setopdefi*
*ned as follows:
- B(T) = w |w|2+, where w ranges over vertices of T.
8 DEV P. SINHA
- B(T ! fln), where fln is a corolla, is the function
` ' ` ' ` ` '
(i, j) 2 n2 7! (Jv(i), Jv(j)) 2 |v|2 |w| ,
w2T 2 +
where v is the join of leaves i and j.
With our choice of definitions, it is straightforward to verify that B is an *
*operad.
Theorem 3.3. The cosimplicial object in Setopassociated to the binomial operad *
*Bo is naturally a sim-
plicial set which is isomorphic to S2o, the simplicial model for S2.
Proof.Recall that S2o= 2o=@ 2o, where 2ois the standard simplicial model for *
* 2. The set n-simplices
of 2 is the set of (x0 x1 . . .xn) 2 {0, 1, 2}n+1, so the cardinality of *
*2nis the (n + 1)st triangular
number. The ith face and degeneracy maps are defined by deleting and repeating *
*xi, respectively. To
obtain S2owe identify all n-tuples in which one of {0, 1, 2} does not appear to*
* a single simplex in each
degree, which is degenerate in positive degrees.
The nth entry of S2ois isomorphic to n2+, the set of unordered pairs of poin*
*ts in n, along with a disjoint
point + which is the image of @ 2ounder the quotient map. The isomorphism recor*
*ds the indices j and
k for which xj-1< xj and xk-1 < xk, when there are two such indices. When there*
* are not two such
indices, such a sequence is identified with the degenerate point +. Under this *
*isomorphism disends + to
+ and for i 6= 0, n sends
( (
(1) (j, k) 7! (ffii(j), ffii(k)) ifffii(j)w6=hffii(k)ereffii(j) = j ifj*
* i
+ otherwise j - 1*
* ifj > i.
For i = 0 and i = n the basic formula is the same, but the (j, k) which get sen*
*t to + are those with j = 1
or k = n, respectively. Similarly, sisends + to + and sends (j, k) 7! (oei(j), *
*oei(k)) where oei(j) = j if j i
or j + 1 otherwise.
By definition Bn = n2+, and it is straightforward to check that the structur*
*e maps of S2oand the
associated cosimplicial object of Bo coincide. To give an example, we unravel t*
*he definition of di with
*
* i
0 < i < n for Bn. These coface maps are given by composites (B(fln) _ +)id_~-!B*
*( n) -Oi!B(fln+1),
i
where n denotes the tree with n root edges and one trivalent internal vertex*
*, which is terminal for
the ith root edge. In Setopthe morphism id _ ~ corresponds to the collapse map*
* in Setwhich sends
2 n 2 n
2 + 2+ _ 2+ to the base point and is the identity on 2+ . The morphism Oi *
*sends (i, i + 1) to
(1, 2) 2 22and sends all other (j, k) to (ffii(j), ffii(k)) 2 n2. The composi*
*te of these two maps coincides
with the definition of difor S2o, as in Equation 1.
The composite of Bo : ! Setopwith X- : Setop! Topgives rise to a functor wh*
*ich we call XBo, for
which the axioms of an operad are immediate to verify. Theorem 3.3 implies the *
*following.
Corollary 3.4. For any Xoin a symmetric monoidal category C, XS2ocanonically de*
*fines an operad through
its isomorphism with XB .
We have yet to find any familiar interpretation for algebras2over these opera*
*d in the categories of spaces
and vector spaces. For spaces the operad structure on XSo does have consequenc*
*es, as we explain in
Example 7.4.
4.The Kontsevich operad
In this section we define an operad structure on the completion of configurat*
*ions in Euclidean space
up translation and scaling defined by Kontsevich [21]. The fact that one could*
* define operads using
OPERADS AND KNOT SPACES *
* 9
the canonical completion of configuration spaces was noticed by Getzler and Jon*
*es [13] soon after this
completion was introduced by Fulton-MacPherson [10] and Axelrod-Singer [1]. Thi*
*s operad structure was
fully developed by Markl [23]. The variant with which we work was first propose*
*d by Kontsevich [21], but
Gaiffi [11] first pointed out the difference with the canonical completion. Ind*
*eed, while Kontsevich called
the following the Fulton-MacPherson operad, we call it the Kontsevich operad to*
* highlight the difference
between the two constructions. Though this construction lacks some of the prope*
*rties of the canoncial
completion, in particular smoothness, is has diagonal and projection maps which*
* satisfy simplicial identities
exactly rather than up to homotopy. These properties led to this construction's*
* independent discovery, its
use, and its naming as the simplicial variant in [34].
We start by setting notation for products of spaces and maps, which we will u*
*se extensively.
Notation. If S is a finite set, XS is the product X#S where #S is the cardinali*
*tyQof S. For coordinates we
use (xs)s2Sor just (xs) when S is understood. Similarly, a product of maps s2S*
*fsmay be written(fs)s2S
or just (fs). Recall that n = {1, . .,.n}.
Definition 4.1. oLet Cn(Rm ) denote the space of (xi) 2 (Rm )n such that if*
* i 6= j then xi6= xj.
Let eCn(Rm ) be the quotient of Cn(Rm ) by the equivalence relation gener*
*ated by translating all of
the xiby some v or multiplying them all by the same positive scalar.
oFor any v 2 Rm - 0, let u(v) = _v_||v||, the unit vector in the direction*
* of v.
*
* n
oLet eCn<[Rm ]> be the closure of the image of eCn(Rm ) under the map (ßij*
*) to (Sm-1)(2), where ßij
sends the equivalence class of (xi) to u(xi- xj).
Note that (ßij) is not injective - it fails to be so on configurations in whi*
*ch all the xilie on some line
- so eCn(Rm ) is not a subspace of eCn<[Rm ]>. But we do have the following th*
*eorem, a consequence of
Corollary 4.5, Lemma 4.12 and Corollary 5.10 of [35].
Theorem 4.2. The canonical map eCn(Rm ) ! eCn<[Rm ]> is a homotopy equivalence.
*
* n
What makes eCn<[Rm ]> manageable is that we can characterize it as a subspace*
* of (Sm-1)(2). It will be
n
convenient to extend coordinates for (uij) 2 (Sm-1)(2)by letting ujibe -uijwhen*
* j > i.
Definition 4.3. oA chain, or k-chain, in S is a collection {i1i2, i2i3, . .*
* . , ik-1ik}, with all ij 2 S
and ij 6= ij+1. Such indices label the edges of a path in the complete gr*
*aph on S. A chain is a
loop, or k-loop, if ik = i1. A chain is straight if it does not contain a*
*ny loops. The reversal of a
chain is the chain ikik-1,n. .,.i2i1.
oA point (uij) 2 (Sm-1)(2)is three-dependentPif for any 3-loop L in n ther*
*e exist aij 0, with at
least one non-zero, such that ij2Laijuij= 0.
oIf S has four elements and is ordered we may associate to a straight 3-ch*
*ain C = {ij, jk, k`} a
permutation of S denoted oe(C) which orders (i, j, k, `). A complementary*
* 3-chain C* is a chain,
unique up to reversal,nwhich is comprised of the three pairs of indices n*
*ot in C.
oA point in (Sm-1)(2)is four-consistent if for any S n of cardinality fo*
*ur and any v, w 2 Sm-1
we have that 0 10 1
X Y Y
(2) (-1)|oe(C)|@ uij. vA@ uij. wA= 0,
C2C3(S) ij2C ij2C*
where C3S is the set of straight 3-chains in S modulo reversal and |oe(C)*
*|is the sign of oe(C).
Points in the image of Cn(Rm ) under (ßij) are three-dependent and four-consi*
*stent, and also satisfy
uij= -uji, a condition we refer to as anti-symmetry. These properties also hold*
* for Cn<[Rm ]>, the closure,
by continuity. Adding the converse, we have the following, which is Theorem 5.1*
*4 of [35].
10 DEV P. SINHA
*
* n
Theorem 4.4. eCn<[Rm ]> is the subspace of all three-dependent, four-consistent*
* points in (Sm-1)(2).
*
* n
We will define operad maps on the completions eCn<[Rm ]> through coordinates *
*of (Sm-1)(2). Embed
n (n)
(Sm-1)(2), and thus eCn<[Rm ]>, in (Sm-1) 2+ as the subspace of (uij)xu+ with u*
*+ equal to the basepoint
of Sm-1, which we choose to be the south pole *S = (0, . .,.0, -1).
Theorem 4.5. The operad structure on (Sm-1)Bo restricts to the subspaces eCn<[R*
*m ]>.
We call the resulting operad, whose nth entry is eCn<[Rm ]>, the Kontsevich o*
*perad Km .
*
* |v|
Proof.Given a tree T, let (uvk`) be a point in (Sm-1)B(T), where v ranges over *
*vertices of T and k, ` 2 2 .
By Definition 3.1, the operad structure on (Sm-1)Bo sends the morphism T ! fln *
*to the map given in
coordinates by (uvk`) 7! (wij)i,j2(n2), where wij= uvJv(i),Jv(j)and v is the jo*
*in vertex of the leaves i and j.
We verify that if the (uvk`) satisfy three-dependence and four-consistency fo*
*r each v, then so does (wij).
For three-dependence, given some wij, wjkand wki, there are two cases to consid*
*er. In the first case the
join in T of leaves i and j lies over that of i and k, so that wjk= -wkior 0wij*
*+ 1wjk+ 1wki= 0. In
the second case the joins of i and j and k are all equal to the same v, so that*
* the dependence of wij, wjk
and wkifollows from that of (uvJv(i)Jv(j)), (uvJv(j)Jv(k)) and (uvJv(k)Jv(i)). *
*Four-consistency works similarly.
Given indices i, j, k and ` the pairwise joins could all equal some v, in which*
* case four consistency of these
{wij} follows from that of {uvk`}. Or, if for example the join of i and j lies *
*over those of i, k and `, then
wik= wjkand wi`= wj`, so four-consistency will follow by the canceling of terms*
* which agree but for
opposite signs.
In [35], we stratify eCn<[Rm ]>, and in particular the points added in closur*
*e. We will not need this
stratification explicitly for our applications, but the related geometry is hel*
*pful in understanding the
operad structure of Km . The stratificationiis indexedjby rp-trees with no red*
*undant edges, with the
v2V (T)
Tth stratum being the image of a map from gC|v|(Rm ) to eCn<[Rm ]> sending (*
*xvi) 7! (uij) with
uij= ßJv(i)Jv(j)((xi)v). Studying this stratification helped lead us to the def*
*inition of the binomial operad.
See section 3 and Theorem 5.14 of [35] for a full development of this geometry,*
* which is illustrated in
Figure 4.6.
Figure 4.6. The effect of an operad structure map associated to the morphism *
* .
The standard completions fCn[Rm ] also constitute entries of an operad, which*
* has been more intensively
studied [13, 21, 23]. The reason we use Km is the following.
OPERADS AND KNOT SPACES *
* 11
Proposition 4.7. The associative operad maps to the Kontsevich operad, for defi*
*niteness by choosing the
basepoint (xij) 2 fCn<[Rm ]> with all xij= *S, for all n.
Finally, we give a comparison between the little disks operad, which we need *
*to formalize, and the
Kontsevich operad.
Definition 4.8. oRecall that the space of n little disks in Dm , the unit d*
*isk, denoted Dm (n) is the
subspace of Cn(Dm ) x (0, 1]n of (xi) x (ri) such that the balls B(xi, ri*
*) are contained in Dm and
have disjoint interiors.
oLet T be a tree whose vertices consist of the root vertex v0 and a termin*
*al vertex ve for each root
edge e. Thus, T ! fln, where n is the number of leaves of T, gives rise t*
*o one of May's structure
maps as in Figure 2.13. Given a label i 2 n let v(i) be the initial verte*
*x for the ith leaf, let o(i)
be the label of leaf i within the ordering on edges of v(i) and let e(i) *
*be the label of the root edge
for which v(i) is terminal.
oDefine Dm (T ! fln) as follows
(xvi, rvi)v2V1(T)7i!#v(yj, æj)j2n where yj= xv0e(j)+ rv0e(j)xv(j)o(j)a*
*ndæj= rv0e(j)rv(j)o(j).
Boardman and Vogt [4] and May [25] showed that algebras over Dm are m-fold lo*
*op spaces.
Theorem 4.9. Let T be a tree with a vertex for each root edge as in Definition *
*4.8 above. The following
diagram commutes up to homotopy,
m(T!fln)
Dm (T)D--------!Dm (n)
? ?
pT?y pn?y
m(T!fln)
Km (T)-K------!Km (n),
where the vertical maps pT are the products of projections pn : Dm (n) ! Cn(Rm *
*) composed with the
canonical maps Cn(Rm ) ! eCn(Rm ) ! eCn<[Rm ]>. Moreover, the vertical maps are*
* homotopy equivalences.
Proof.We define the homotopy explicitly. Define H : Dm (T) x (0, 1] ! eCn(Rm ) *
*by sending (xvi, rvi) as
in Definition 4.8 and t 2 (0, 1] to the coset of (yj(t)) with yj(t) = xv0e(j)+ *
*t . rv0e(j)xv(j)o(j). We claim that
__ __
H extends uniquely to H : Dm (T) x [0, 1] ! eCn<[Rm ]>, and that H coincides wi*
*th Km (T ! fln) O pT
when t = 1. Consider uij= ßij((yk(t))). If the join of leaves i and j is one of*
* the non-root vertices, so
v(i) = v(j), then uijwill be equal to the unit vector in the direction of xv(j)*
*o(j)- xv(j)o(i), independent of t. If
the join of leaves i and j is the root vertex, then as t approaches 0, uijappro*
*aches the unit vector in the
direction of xv0e(j)- xv0e(i). These limiting values coincide with the definiti*
*on of Km (T ! fln)(xvi).
That the projection Dm (n) ! Cn(Dm ) is a homotopy equivalence is standard, k*
*nown since the definition
of little cubes in [3], so by Theorem 4.2 the maps pn are homotopy equivalences.
In fact, [21] claims that these Dm and Km are homotopy equivalent operads, wh*
*ich we assume to mean
that there is a chain of equivalences of maps of operads, that is maps which co*
*mmute with structure maps
exactly. We will not need this stronger claim. Recall that the homology of an o*
*perad of spaces with field
coefficients is an operad of vector spaces by the Künneth theorem. The homology*
* of the little disks operad
has a well-known description.
Definition 4.10. The kth entry of the degree n Poisson operad Poissn(k) is the *
*submodule of the sym-
metric algebra on the free graded Lie algebra over k variables x1, . .,.xk span*
*ned by monomials in which
all variables appear exactly once. Monomials are graded by putting all xiin deg*
*ree zero and giving the
bracket degree n. So for example [x1, x3] [[x4, x2]x5]and x1x2. .x.5are element*
*s of Poiss3(5) of degree
nine and zero respectively.
12 DEV P. SINHA
The map Oi : Poissn(j) Poissn(k) ! Poissn(j + k - 1) sends m1 m2 to the mo*
*nomial defined as
follows.
oFor each j, substitute xj+i-1for xj in m2 to obtain ___m2.
oIn m1, substitute xj+i-1for xj if j > i and ___m2for xito obtain m.
oReduce m according to the graded Leibniz rule
[a, bc] = [a, b]c + (-1)(|a|+n+1)|b|b[a, c],
to obtain an element of Poissn(i + j - 1).
The following corollary is essentially a summary of Fred Cohen's famous calcu*
*lation of the homology of
Dm [9]. We also plan to give an exposition of this result in [36].
Theorem 4.11. The homology of Km is the degree m - 1, Poisson operad Poissm-1.
5.Models for spaces of knots and immersions arising from the calculus of embed*
*dings
5.1. A brief overview of the calculus of embeddings. Our main theorems connect *
*the theory of
operads to Goodwillie calculus. We first informally introduce some terminology*
* from the calculus of
embeddings (see Weiss's [43] for an excellent introduction and [44] for a full *
*treatment), and then precisely
state the theorems we use. The main spaces with whom we are concerned are relat*
*ed to embeddings and
immersions.
Definition 5.1. Let Emb(M, N) denote the space of embeddings of M in N, topolog*
*ized as a subspace
of the space of all maps, with the compact-open topology. Similarly, let Imm (*
*M, N) be the space of
immersions of M in N. If M and N have boundary, we usually specify some boundar*
*y conditions. In
particular, if M = I and N = Im , we let *+ = (0, . .,.0, 1) 2 Im , *- = (0, . *
*.,.0, -1) and demand that the
endpoints of I map to *+ and *- with tangent vectors *S.
By results of Palais [30], these spaces are dominated by simplicial complexes*
* and thus homotopy equiv-
alent to CW-complexes [29].
In the calculus of embeddings, one views spaces of embeddings, immersions and*
* other moduli in dif-
ferential topology as functors from the poset of open subsets of M to topologic*
*al spaces, a philosophy
originally due to Gromov. Ultimately interested in the value of the functor at *
*the open set which is all of
M, one tries to use homotopy limits to interpolate that value from values of th*
*e functor at simple open
sets, namely those which are diffeomorphic to a union of open balls. Functors f*
*or which interpolation using
a finite number of balls works perfectly are called polynomial, and those for w*
*hich interpolation works in
the limit as the number of balls tends to infinity are called analytic. Weiss s*
*hows in [44] that polynomial
functors are those which satisfy higher-order Mayer-Vietoris conditions, and Go*
*odwillie-Weiss show in [15]
that analyticity follows from satisfying those conditions through an increasing*
* range of connectivity. More
formally we have the following.
Definition 5.2. oFor any manifold W of dimension m let U(M) be the category*
* of open subsets
of M under inclusion, and let Uk(M) be the full sub-category of U(M) of o*
*pen sets diffeomorphic
to tiRm , where i k.
oFor any contravariant functor F from U(M) to spaces let TkF be the functo*
*r which sends W to
holimU-2Uk(W)F(U).
oLet bk(F) : TkF ! Tk-1F be the canonical natural transformation defined b*
*y restricting Uk(M)
to Uk-1(M), and let T1 F be the homotopy inverse limit of the TkF over th*
*ese restrictions.
oLet jk(F) : F ! TkF be the canonical natural transformation arising from *
*the maps F(M) !
F(U) for U 2 Uk(M). If by context F is understood, we will use the simple*
*r notation jk.
oThe natural transformations jk commute with the bk, so let j1 : F ! T1 F *
*be the limiting natural
transformation.
OPERADS AND KNOT SPACES *
* 13
The sequence T0F b1T1F b2T2F . .i.s called the Taylor tower for F. Analytic*
*ity means that the
homotopy inverse limit of this tower is weakly equivalent to F. The motivating *
*example for this circle of
ideas is that of immersions.
Theorem 5.3. If dim(M) < dim(N) then for k 1, jk : Imm(U, N) ! TkImm (U, N) i*
*s a weak equiva-
lence for any U M.
This theorem follows from Example 2.3 of [44], which says that immersions are*
* a linear functor, and
the commentary after Theorem 5.1 of [44]. The embedding functor is not polynomi*
*al but by theorems of
Goodwillie, Klein and Weiss it is analytic. The following Theorem is essentiall*
*y Corollary 2.5 of [15].
Theorem 5.4. If dim(M) < dim(N) - 2 then j1 (Emb ) is a weak equivalence.
For dim(M) < dim(N) - 2 as stated, this theorem requires deep disjunction res*
*ults of Goodwillie,
and surgery results of Goodwillie-Klein [19, 18]. If dim(M) is less than roughl*
*y dim(N)_2there are much
easier methods, using only the Blakers-Massey theorem and dimension counting, f*
*or proving the needed
higher-order Mayer-Vietoris conditions.
5.2. Knot space models through homotopy limits of configuration spaces. Definit*
*ion 5.2 of
TkEmb (M, N) is ephemeral, but the building blocks, namely spaces of embeddings*
* of balls, are essen-
tially configuration spaces. Goodwillie, Klein and Weiss have given more concre*
*te models for the spaces
in this Taylor tower (or for the homotopy fibers of Tk ! Tk-1, which are called*
* layers), either as spaces of
sections, as in section nine of [44], or as mapping spaces with strongly define*
*d equivariance properties, as
in [16]. In the case of knots we have developed three closely-related models fo*
*r these polynomial approxi-
mations [34] and used them for both computational and geometric applications [3*
*2, 8]. These models all
utilize completions of configuration spaces constructed similarly to the Kontse*
*vich operad.
n
Definition 5.5. oLet An<[Im ]> be the product (Im )n x (Sm-1)(2), with coor*
*dinates (xi) x (uij).
oLet Cn<[Im ]> be the closure of the image of Cn(Im ) under ' x (ßij), whe*
*re ' is the inclusion of
Cn(Im ) in (Im )n.
oLet Cn<[Im , @]> be the closure in Cn+2<[Im ]> of the subspace of Cn+2(Im*
* ) with x1 = *+ =
(0, . .,.0, 1) and xn+2= *-.
In [35] we study Cn<[Im ]> by relating it to the canonical compactification C*
*n[Im ], which is a manifold
with corners. We characterize Cn<[Im ]> as a subspace of its defining ambient s*
*pace, as stated for eCn<[Rk]>
in Theorem 4.4. The following is essentially Theorem 5.14 of [35].
Theorem 5.6. Cn<[Ik]> is the subspace of (xi) x (uij) such that (uij) 2 eCn<[Rk*
*]> and if xi6= xj then uij
is u(xj- xi).
In our models, we need diagonal maps between configuration spaces. The idea *
*is to add a point
"infinitesimally far" from one point in a configuration, but to do so entails c*
*hoosing a unit tangent vector
at that point.
*
* n
Definition 5.7. Let C0n<[Im ]> = Cn<[Im ]> x (Sm-1)n. Let A0n<[Im ]> = (Im x Sm*
*-1)n x (Sm-1)(2), which
is canonically diffeomorphic to (Im )n x (Sm-1)n2. We use coordinates for this *
*latter presentation of the
form (xi) x (uij) with i and j possibly equal.
As in Theorem 4.5 we define maps between the C0n<[Im ]> at the level of the a*
*mbient spaces A0n<[Im ]>,
using Theorem 5.6 to check that they restrict appropriately. We are aided by th*
*e following combinatorial
shorthand.
Definition 5.8. oGiven a map of sets oe : R ! S let pXoeor just poedenote t*
*he map from XS to XR
which sends (xi)i2Sto (xoe(j))j2R.
14 DEV P. SINHA
oGiven oe : m ! n, define Aoe: A0n<[Im ]> ! A0m<[Im ]> as pImoex pSk-1oe2.
Proposition 5.9 (Proposition 6.6 of [35]). The restriciton Aoeto C0n<[Im ]> map*
*s to C0m<[Im ]>. If oe sends
1 ! 1 and n ! m then Aoealso restricts to a map, which we call Foe, from C0n-2<*
*[I, @]> to C0m-2<[I, @]>.
We may now define diagonal maps on compactified configuration spaces with tan*
*gential data.
Definition 5.10. Let ffii: C0n<[Im , @]> ! C0n+1<[Im , @]> be Foeiwhere oei: n_*
*+_3_! n_+_2_sends j to itself if
j i or j - 1 if j > i.
A final key property of this compactification is that it is functorial for em*
*beddings. The proof of the
following theorem is identical to that of Corollary 4.8 of [35], using Theorem *
*5.8 of [35] and the fact that
Cn<[I]> = n. Recall that for a nonzero vector v 2 Rm , u(v) = _v_||v||.
Theorem 5.11. For an embedding f : I ! Ik there is an evaluation map evn(f) : *
*n ! Cn<[Ik]> which
extends the map from the interior of n to C0n(Ik) sending (ti) to (f(ti)) x (u*
*(f0(ti))).
One of the main themes of [34] is connecting this evaluation map with the cal*
*culus of embeddings. Ap-
plying this calculus to embeddings of the unit interval is simpler than to embe*
*ddings of higher-dimensional
manifolds because the category Uk(I) may be replaced by the category of subsets*
* of a finite set (see
Definition 2.6).
Definition 5.12. Let Dmkbe the functor from P0(k) to spaces which sends S [k]*
* to C0#S-1<[Im , @]> and
which sends the inclusion S S [ j to the map ffiiwhere i is the number of ele*
*ments of S less than j. Let
Dmk= holimD-mk.
In the notation of [34], Dmkwould be Dk<[Im ]>. Because the realization of P0*
*(k) is k and all of the maps
ffii are inclusions of subspaces, Dmkis a subspace of Maps( k, C0k<[Im , @]>). *
*If f 2 Emb(I, Im ) is a knot,
evk(f) defines an element of Dmk, as we may simply check that if tj= tj+1for so*
*me point (ti) 2 k then
the image of evk(f) ((ti))is in the image of ffij. By abuse, let evk denote the*
* adjoint map from Emb(I, Im )
to Dmk. Building on the simpler üc tting method" definition of TkEmb (I, Im ), *
*as described in Section 3 of
[34], Lemma 5.18 and the proof of Theorem 5.3 of [34] establish the following.
Theorem 5.13. Dmkis homotopy equivalent to TkEmb (I, Im ), and evk agrees with *
*jk in the homotopy
category.
Though not historically expressed in these terms, there is a similar theorem *
*for immersions of an interval.
Definition 5.14. oLet di : (Sm-1)j ! (Sm-1)j+1be the ith diagonal inclusion*
*, which by con-
vention for i = 0 and i = j + 1 are insertion of the basepoint *S as the *
*first, respectively last,
coordinate.
oLet Gmkbe the functor from P0(k) to spaces which sends S [k] to (Sm-1)#*
*S-1 and which sends
the inclusion S S [ j to the diagonal map diwhere i is the number of el*
*ements of S less than j.
oLet Gmk= holimG-mk.
As was true for Dmk, Gmkis a subspace of the space of maps from k to the ter*
*minal space of Gmk, namely
(Sm-1)k. The evaluation map for immersions is the unit derivative map. By abuse*
*, let evk : Imm(I, Im ) !
Gmksend f to the map which sends t1, . .,.tk to uf0(t1), . .,.uf0(tk).
Theorem 5.15. If k 1, Gmkis homotopy equivalent to TkImm (I, Im ), and thus t*
*o Imm(I, Im ). Moreover,
evk agrees with jk in the homotopy category.
Sketch of proof.There are many ways to establish this theorem. By the Hirsh-Sm*
*ale theorem [37],
Imm (I, Im ) is homotopy equivalent to Sm-1, through the unit derivative map. *
*But ev1 : Imm(I, Im ) !
holim(-* ! Sm-1 *)is also the unit derivative map, which establishes the theo*
*rem for k = 1. For the
OPERADS AND KNOT SPACES *
* 15
other k, we may use Lemma 2.8, since Gmkis the pull-back of the standard cosimp*
*licial model for Sm-1
through the functor ck of Definition 2.7. The kth totalization of this cosimpli*
*cial model, which is fibrant,
is homeomorphic to Sm-1 if k 1, from which it follows that Gmkis homotopy eq*
*uivalent to Sm-1. The
map from Sm-1 to the kth totalization, and thus Gmk, is through evaluation of *
*the unit derivative.
Let ø : Emb(I, Im ) ! Imm(I, Im ) denote the inclusion. Let æmk: Dmk! Gmkdeno*
*te the map of diagrams
defined on each entry by projection from C0n<[Im , @]> = Cn<[Im , @]> x (Sm-1)n*
* onto (Sm-1)n, and let pmk
also denote the induced map on homotopy limits.
Proposition 5.16. The square
Emb (I, Im-)ø---!Imm(I, Im )
? ?
evk?y evk?y
m
Dmk -pk---! Gmk.
commutes. Moreover, pmkagrees with Tk(ø) in the homotopy category.
Sketch of proof.The commutativity of the diagram is immediate from the definiti*
*ons. That pmkagrees
with Tk(ø) in the homotopy category ultimately follows from the fact that for U*
* a disjoint union of k + 2
open intervals, two of which contain endpoints of I and thus are fixed at one e*
*nd, we have Emb(U, Im ) '
C0k<[Im , @]>, Imm (U, Im ) ' (Sm-1)k and the inclusions from embeddings to imm*
*ersions coincides with
projection, as in the definition of pmk.
5.3. A closer look at Em .
Proposition 5.17. The inclusion ø : Emb(I, Im ) ! Imm(I, Im ) is null-homotopic*
*, so
Em ' Emb(I, Im ) x Imm (I, Im ) ' Emb(I, Im ) x 2Sm-1.
Proof.Given f 2 Emb(I, Im ) consider the map æ(f) : 2 ! Sm-1 which sends t1, t*
*2 to either u(f(t2) -
f(t1)) if t1 6= t2 or u(f0(t)) if t1 = t2 = t. We may view æ(f) as a homotopy b*
*etween ev1(f), which is the
restriction to the t1 = t2 edge, and the restriction to the t1 = 0 and t2 = 1 e*
*dges. But the restriction to
these latter two edges is canonically null-homotopic, since their images lie in*
* the southern hemisphere of
Sm-1. Thus, ev1 restricted to Emb(I, Im ) is null-homotopic. Since ev1 is an eq*
*uivalence on Imm(I, Im )
this implies that the inclusion of Emb(I, Im ) is null-homotopic.
That Em ' Emb(I, Im ) x Imm (I, Im ) is immediate from its definition as the*
* homotopy fiber of this
inclusion, and that this is in turn weakly equivalent to Emb(I, Im )x 2Sm-1 fol*
*lows from the Hirsch-Smale
theorem [37].
The embedding space Em is related to the space of framed knots when m = 3. Be*
*cause the normal
bundle of an embedded interval in Im is essentially an oriented bundle over S1,*
* it is trivial. By fixing
one framing of the normal bundle, all others are related to it by a map from I *
*to SO(m - 1) fixed at
the endpoints of I. The space of framed knots is thus homotopy equivalent to Em*
*b(I, Im ) x SO(m - 1).
For m = 3 we get Emb (I, I3) x S1, which is homotopy equivalent to Emb (I, I3)*
* x Z. There is thus a
suspension map j on the second factor to Emb(I, I3) x 2S2 ' E3.
Proposition 5.18. The suspension map j from the space of framed knots in I3 to *
*E3 is a bijection on
components.
Framings naturally arise in defining knot invariants through our operad model*
* because of this bijection.
We also see 2S3 as the homotopy fiber of j, which might give a new viewpoint o*
*n the results of [2].
16 DEV P. SINHA
6.The main result
We assemble our work to this point to prove the main result. As needed for th*
*e calculus of functors,
extend Em to be a functor on the open sets of I by sending U to the homotopy fi*
*ber of the inclusion
Emb (U, Im ) ! Imm(U, Im ). We will recover models for Em from those for embedd*
*ings and immersion
spaces.
Lemma 6.1. If A and B are two functors from U(M) to spaces with a natural trans*
*formation ø between
them, and F is defined by F(U) = hofib(ø : A(U) ! B(U)), then Tk(F) = hofib(Tk(*
*A) ! Tk(B)).
Proof.The equality is immediate from the definition of Tk, since taking homotop*
*y fibers commutes with
taking homotopy limits.
In defining a fiber to æmkwe are led to the following.
Definition 6.2. Let ei: An<[Im ]> ! An+1<[Im ]> send (uj`) to (vj`) where vi,i+*
*1= *S, the basepoint of
Sm-1 and other vj`are equal to uoei(j)oei(`), where as before oei(j) = j or j -*
* 1 if j < i or j > i respectively.
By abuse, also use ei to denote its restriction to Cn<[Im , @]> mapping to Cn+1*
*<[Im , @]> as one can check
using Theorem 5.6.
Alternately, ei: Cn<[Im , @]> ! Cn+1<[Im , @]> is the restriction of ffiito C*
*n<[Im , @]> x (*S)n C0n<[Im , @]>.
Definition 6.3. Let Fmkbe the functor from P0(k) to spaces which sends S [k] *
*to C#S-1<[Im , @]> and
which sends the inclusion S S [ j to the map eiwhere i is the number of eleme*
*nts of S less than j. Let
Fmk= holimF-mk.
Theorem 6.4. Fmk is homotopy equivalent to TkEm . For m > 3, j1 : Em ! holimT-k*
*Emis a weak
equivalence.
Proof.We use the models Dmkand Gmkfor TkEmb and TkImm as given in Theorems 5.13*
* and 5.15 respec-
tively. By Proposition 5.16, pmk: Dmk! Gmkagrees with Tk of the inclusion from *
*embeddings to immersions.
Applying Lemma 6.1 with A = Emb(-, Im ), B = Imm(-, Im ), and the natural trans*
*formation between
them be the standard inclusion, we have that TkEm = hofibpmk.
If a map diagrams indexed by P0(k) is a fibration object-wise, then the induc*
*ed map on homotopy
limits is a fibration and the fiber is given by the homotopy limit of the fiber*
*s object-wise (see for example
Lemma 3.5 of [8]). Because æmkis a fibration object-wise, we thus identify hofi*
*bpmk= hofib(holimæ-mk)
with such a homotopy limit of object-wise fibers. By our definition, the diagra*
*m of fibers is Fmk, whose
homotopy limit is Fmk, establishing the first half of the theorem.
The second half of the theorem is immediate from Theorems 5.3 and 5.4.
Note that because of Proposition 5.17, we could extend Em to a functor on U(I*
*) by setting E!m(U) =
Emb (U, Im )x Imm (U, Im ). The extension E!mwould lead to a set of approximati*
*ons to Em different from
the Fmk. Also, while Em decomposes as a product in Proposition 5.17, at the lev*
*el of entries of diagrams
it is in the approximation to Emb(I, Im ) that we see a product.
Recall Proposition 4.7 that Km is an operad with multiplication, which using *
*Definition 2.16 has an
associated cosimplicial object. We translate from Fmkto gTot(Kom), essentially *
*through the standard pro-
jection Cn<[Im , @]> ! fCn<[Rm ]>. We modify both Cn<[Im , @]> and this projec*
*tion to define a natural
transformation.
Definition 6.5. oLet " <= 1_6. For x 2 Rm , let d+(x) be the distance in R*
*m from x to *+ =
(0, . .,.0, 1) and d-(x) be the distance to *-. Let flj : Rm ! R be proj*
*ection onto the jth
coordinate.
OPERADS AND KNOT SPACES *
* 17
oLet Cn<[Im , @"]> be the subspace of (xi) x (uij) 2 Cn<[Im , @]> where if*
* d+(xi) and d+(xj) are less
than ä nd i < j then flk(xi) = flk(xj) for k < m and flm (xi) flm (xj)*
*. Moreover, if xi= xj and
i < j then uij= *S.
oLet Fmk,"be the functor from P0(k) to spaces which sends S [k] to C#S-1*
*<[Im , @"]> and which
sends the inclusion S S [ j to the map eiwhere i is the number of eleme*
*nts of S less than j.
oLet Fmk,"= holimF-mk,".
Proposition 6.6. The map Fmk,"! Fmk, induced by the natural transformation ' : *
*Fmk,"! Fmkwhich at
each entry is the canonical inclusion, is a homotopy equivalence.
Proof.It suffices to check ' is a homotopy equivalence object-wise, for which w*
*e adapt the machinery
developed in [35] for compactified configuration spaces. Both Ck<[Im , @"]> and*
* Ck<[Im , @]> are quotients
of the canonical Axelrod-Singer compactifications which we call Ck[Im , @"] and*
* Ck[Im , @] respectively; see
Definitions 1.3 and 4.18 of [35] for the definition of Ck[Im , @], which can be*
* modified as in Definition 6.5 for
Ck[Im , @"]. These quotient maps are homotopy equivalences, by the proof of The*
*orem 5.10 of [35], which
applies verbatim in these cases.
Ck[Im , @"] retracts to its subspace Ck[Im -N"], where N" is the union of the*
* " neighborhoods of *+ and
*- by scaling the xiby 1-". Both Ck[Im , @] and Ck[Im -N"] are manifolds with c*
*orners (see Theorem 4.4 of
[35]), and thus are homotopy equivalent to their interiors, Ck(Int(Im )) and Ck*
*(Int(Im -N")) respectively.
But these interior configuration spaces are diffeomorphic, since Int(Im ) and I*
*nt(Im -N") are. Composing
this diffeomorphism with the previous homotopy equivalences establishes the equ*
*ivalence of Ck<[Im , @"]>
and Ck<[Im , @]> and thus establishes the result.
We use Ck<[Im , @"]> because they readily project to fCk<[Rm ]>.
Definition 6.7. oLet (ai)mi=1, ai 2 R denote a point in Rm . Define ~+ : (R*
*m - *+) ! Rm by
sending (ai) to (bi) where if i 6= m then bi= aiand
( ä m
_____ d+(ai) < "
bm = d+(ai)
am d+(ai) ".
Define ~- : (Rm - *-) ! Rm similarly, and let ~ = ~+ O ~-.
k
oDefine ßk : Ck<[Im , @"]> ! fCk<[Rm ]> (Sm-1)(2)by sending (xi) x (uij)*
* to (vij) where vijis:
- u(~(xi) - ~(xj)) if xi6= xj and neither equals *+ or *-.
- The Jacobian on ~ applied to uijif xi= xj.
- *S, if xi6= xj and either xi= *+ or xj= *-.
Proposition 6.8. ßk is continuous.
Proof.We first identify ßk on the subspace t Ck<[Im - (*+ [ *-)]> with the comp*
*osite of Ck<[~]>, the
map on configuration spaces induced by the embedding ~ (see Corollary 4.8 of [3*
*5]), and the canonical
projection Ck<[Rm ]> to fCk<[Rm ]>. What remains is to check continuity on the *
*subspace in which some
xn = *+. Consider a sequence {(x`i), (u`ij)}1`=1with limit point (x1i) x (u1ij)*
*, so that x1n= *+. We show
that its image under ßk has vnj which approaches *S if x1j6= *+ or n < j or whi*
*ch approaches -*S
otherwise. For each j, either x1j2 N"+, which is also true for ` sufficiently l*
*arge, in which case unjmust
be *S if n < j or -*S if n > j, so that the sequence vnjwould be eventually con*
*stant at *S or -*S. Or
if x1j=2N" then as x`n7! *+ the last coordinate of ~+(xi`) becomes arbitrarily *
*large. Because xj`7! xj
stays in Im we have u(~(x`n) - ~(x`j)) 7! *S. Continuity when some xn = *- work*
*s similarly.
We now may assemble our main result, Theorem 1.1, which casts the embedding c*
*alculus tower for Em
in the language of operads. For convenience, we restate the theorem here.
18 DEV P. SINHA
Theorem 6.9. The kth approximation to Em in the embedding calculus, namely TkEm*
* , is weakly equivalent
k
to gTotKom.
Proof.We will check that the maps ßk assemble to a natural transformation of fu*
*nctors from Fmkto KomOck,
with ck as in Definition 2.7, which gives rise to a weak equivalence on homotop*
*y limits. Theorem 6.4 then
says that the homotopy limit of Fmk is weakly equivalent to TkEm , and Lemma 2.*
*8 implies that the
k
homotopy limit of KomO ck is weakly equivalent to gTotKom, establishing the the*
*orem.
For the assembled ßk to be a natural transformation, we must have ßk O ei= di*
*O ßk. For most i this
is immediate to check, as repeating coordinates and passing to the quotient eCk*
*<[Rm ]> are processes which
clearly commute. The i = 0 and i = k + 1 cases require the modifications we mad*
*e in Definition 6.7. For
KomO ck we trace through Definitions 2.16 and 3.1, Theorem 4.5 and Proposition *
*4.7 see that dk+1takes a
point (uij) 2 eCk<[Rm ]> leaves all these uijunchanged and adds ui,k+1= *S for *
*all i to obtain a point in
Cek+1<[Rm ]>. On the other hand, ek+1adds the k + 1st point to the configuratio*
*n at *-, which under ßk
will also lead to all ui,k+1= *S. The i = 0 case works similarly.
The fact that the assembled ßk induce a weak equivalence on homotopy limits f*
*ollows from it being a
homotopy equivalence object-wise. We already know from the proof of Proposition*
* 6.6 that Ck<[Im , @"]>
is homotopy equivalent to the subspace Ck(Int(Im - N")), which is diffeomorphic*
* to Ck(Rm ). Composed
with this diffeomorphism on this subspace, ßk is the standard projection Ck(Rm *
*) ! fCk(Rm ) followed by
the canonical map to fCk<[Rm ]> which is a homotopy equivalence by Corollaries *
*4.5 and 5.9 of [35].
7.Observations and consequences
7.1. Spectral sequences. The results in this section parallel those of section *
*7 of [34]. Applying the
homotopy spectral sequence of Proposition 2.9 for Komwe immediately have the fo*
*llowing.
Theorem 7.1. There is a spectral sequence converging to ß*(gTotKom) with
"
E-p,q1= kersk* ßq(Cp(Rm )).
The d1 differential is the restriction to this kernel of the map
p+1i=0(-1)idi*:ßq(Cp-1(Rm )) ! ßq(Cp(Rm )).
Theorem 6.4 implies that this spectral sequence computes homotopy groups of E*
*m when m 4. Except
for in the p = 1 column, this spectral sequence coincides exactly with that stu*
*died with rational coefficients
in [32], so we do not give a more explicit description here. The rows of this s*
*pectral sequence have also
been examined by Kontsevich [22].
For m = 3, the case of classical knots, we conjecture that jk : Em ! TkEm is *
*a universal type-(k - 1)
framed knot invariant over the integers. For k 3, we may deduce this from the*
* main results of [8]. We
conjecture that the entries E1-k,kof this spectral sequence are isomorphic to t*
*he module of primitive weight
systems of degree k - 1 over the integers, which would be a first step to this *
*conjecture. We have checked
that for small k, the group E2-k,kas described purely algebraically in Theorem *
*7.1 is isomorphic to this
module of primitives, but have not resolved this algebraic question in general.
In light of Theorem 4.11, the homology spectral sequence from Theorem 2.10 ha*
*s a pleasant description.
Recall Definition 2.16, which for operads of vector spaces introduces the notat*
*ion of HH*(O) for the total
cohomology of the associated cosimplicial object.
Theorem 7.2. There is a spectral sequence with E2-p,q= HHp,q(Poissm) which for *
*m 4 converges to
the homology of gTotKom, and thus of Em .
Proof.If we use the second description of the homology spectral sequence from T*
*heorem 2.10, then E1-*,*
will be H*(Kom), which is the Poisson operad by Theorem 4.11. The induced opera*
*d with multiplication
OPERADS AND KNOT SPACES *
* 19
structure on the Poisson operad is the standard one. Thus, the d1 differential*
* will coincide with the
differential for total cohomology of the Poisson operad, and the E2 term will b*
*e the total (or Hochschild)
cohomology of the Poisson operad as stated.
It remains to check the convergence conditions of Theorem 2.10. In the case o*
*f the Kontsevich operad,
the entries Kkm= eCk<[Rm ]> are homotopy equivalent to Ck(Rm ), which are simpl*
*y connected if m 3.
Using the first definition of Theorem 2.10, we start with H*(Cp(Rm )) and expli*
*citly understand the kernels
of the maps si*. We use Theorem 4.11 and Definition 4.10 to identify H*(Cp(Rm )*
*) in terms of products of
brackets in variables x1, . .,.xp. Tracing through the definitions of the assoc*
*iated cosimplicial object, si
sends a product of brackets in the xjto either zero, if the variable xiappears *
*in a bracket, or the monomial
obtained by removing xiand relabeling xj to xj-1for j > 1, if xidoes not appear*
* in a bracket. Therefore
to be in the kernel of all of the si, all of the variables ximust appear in a b*
*racket, so there must be at
least k_2brackets, leading to a total degree of at least k(m-1)_2. For m > 3, t*
*his is greater than k and thus
gives the estimate needed for application of Theorem 2.10.
This spectral sequence in rational cohomology can also be viewed as arising f*
*or the homotopy groups
of the Taylor tower for the functor to spectra U 7! Q ^ Em (U), the rational Ei*
*lenberg-MaClane spectrum
smashed with Em (U). For m = 3, Volic's results [41, 42] imply that the map fro*
*m the knot space to this
Taylor tower serves as a universal framed finite-type invariant over the ration*
*al numbers.
7.2. A little two-cubes action from the McClure-Smith framework. Theorem 1.1 fi*
*ts perfectly
into the framework created by McClure and Smith in their solution of the Delign*
*e conjecture [26]. One of
their central results is the following.
Theorem 7.3. The totalization of the associated cosimplicial object of an opera*
*d with multiplication admits
an action of an operad equivalent to the little 2-cubes operad, as does its hom*
*otopy invariant totalization.
Proof.We are simply collecting results from [26] and [27]. For the standard tot*
*alization, we are simply
quoting Theorem 3.3 in [26]. For the homotopy invariant totalization, Theorem 1*
*5.3 of [27] says that gTot
of any cosimplicial space with what they call a 2-structure has an action of a*
*n operad equivalent to the
little 2-cubes. Proposition 10.3 of [27] identifies an operad with multiplicati*
*on structure on a sequence of
spaces with a 2 structure.
Example 7.4. Consider the cosimplicial model for the space of maps from S2 to X*
*, namely XS2o. By
Theorem 3.3 S2o~=Bo, so there is an operad structurenon this collection of spac*
*es. In order to get an
operad with multiplication, we restrict each XB = (xff) to the subspace in whi*
*ch x+ = *, where + is the
basepoint2of Bn and * is the base point of X. The operad structure maps restric*
*t appropriately, and we
obtain XSo?, to which the associative operad maps2at each level to the point wi*
*th all xff= *.
Applying Theorem 7.3, the totalization of XSo?is a little 2-cubes space, and *
*we know that its totalization
is 2X. McClure and Smith fully develop this example (in fact for all cases, no*
*t just n = 2) in Section 11
of [27]. They show that the little 2-cubes action which arises in this example *
*coincides with the standard
one.
We can immediately establish Theorem 1.4, one of our main results, which for *
*convenience we restate
here.
Theorem 7.5. For any m, there is a little two-cubes action on gTot(Kom). For m *
*> 3, Em is a two-fold
loop space.
Proof.Applying Theorem 7.3 for the Kontsevich operad with its given multiplicat*
*ion establishes the two-
cubes action.
By Theorem 1.1, if m 4, Em is homotopy equivalent to gTot(Kom), so it has a*
* 2-cubes action as well.
But Em is connected for m 4, since both (Imm (I, Im )) ' 2Sm-1 and Emb(I, I*
*m ) are connected (that
20 DEV P. SINHA
the latter space is connected is because any path through maps from an embeddin*
*g to the standard one
becomes an isotopy once put in general position). So by the recognition theorem*
* of [3, 4], Em is a 2-fold
loop space.
We expect this two-cubes action to be important for closer examination of the*
* homotopy type, in
particular the homology, of Em . In [7], Budney constructs a little two-cubes a*
*ction directly on a different
space closely related to Emb(I, Im ), namely the space of framed knots. He goes*
* on to show that the two-
cubes action is free when m = 3, generated by the components of prime knots. He*
* identifies the homotopy
types of a large class of prime components. It would be interesting to see if B*
*udney's two-cubes action is
related to ours on Em , and if there is a two-cubes action to be found on Emb(I*
*, Im ) itself. Perhaps this
two-cubes action respects the product decomposition of Em . There is a possible*
* first step towards such a
result as follows. *
* o
We defined Km , following Theorem 4.5, with entries as subspaces of those in *
*(Sm-1?)B , which is
(Sm-1?)S2oby Theorem 3.3. Let 'o denote the corresponding map of operads with m*
*ultiplication and thus
of associated cosimplicial spaces. We conjecture that the following diagram, in*
* which the bottom arrow is
the standard projection along with the identification given by the Hirsch-Smale*
* Theorem, commutes
o) 2
gTot(Kom) gTot('-----!gTot((Sm-1?)So)
x ?
'?? ~=?y
Emb(I, Im ) x Imm (I,-Im-)--! 2Sm-1.
If it does, then we may be able to use the McClure-Smith machinery to define a *
*two-cubes action on the
homotopy fiber of 'o, whose totalization would be Emb(I, Im ).
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Department of Mathematics, University of Oregon, Eugene, OR 97403