THE TOPOLOGY OF SPACES OF KNOTS
DEV P. SINHA
Abstract.We present two models for the space of knots which have endpoint*
*s at fixed boundary points
in a manifold with boundary, one model defined as an inverse limit of map*
*ping spaces and another which is
cosimplicial. These models are homotopy equivalent to the corresponding k*
*not spaces when the dimension
of the ambient manifold is greater than three, and there are spectral seq*
*uences with identifiable E1terms
which converge to their cohomology and homotopy groups. The combinatorics*
* of the spectral sequences
is comparable to combinatorics which arises in finite-type invariant theo*
*ry.
Contents
1. Introduction 1
1.1. Basic definitions, notation and conventions *
* 2
1.2. Acknowledgments 3
2. Heuristic understanding of the mapping space and cosimplicial models *
* 3
3. Goodwillie's cutting method *
* 4
3.1. Homotopy limits 5
3.2. Polynomial approximations to spaces of knots *
* 6
4. Fulton-MacPherson compactifications and categories of trees *
* 7
4.1. Cn[M], the basic compactification and n, the basic category of trees *
* 7
4.2. Cn[M, @] for manifolds with boundary, associahedra, and planar trees *
* 10
4.3. Cn<[M]> the öc simplicial" variant, tangential data, and diagonal maps *
* 12
5. The mapping space model *
*15
6. The cosimplicial model *
* 19
7. The cohomology of ordered configurations in Euclidean space *
* 22
8. The spectral sequences *
* 23
9. Further work and open questions *
* 25
References 27
1.Introduction
In this paper, we study the topology of spaces of embeddings of one-dimension*
*al manifolds, Emb(I, M),
where M is a smooth manifold with boundary and the embeddings must send the end*
*-points of I to fixed
boundary points of M with specified tangent vectors at those points. These spac*
*es have gained considerable
attention recently, especially in the work of Vassiliev [35] and Kontsevich [22*
*, 23]. Also, Hatcher has been
studying the topology of various components of the space of knots in R3 [20].
We build on the approach of Goodwillie and his collaborators [18, 17, 16, 36]*
*, which is known as the
calculus of embeddings (or more generally of isotopy functors). Their approach *
*produces a äT ylor Tower"
of öp lynomial approximations" to embedding spaces. From the spaces in this tow*
*er we produce two new
models. The first, which we call a mapping space model, is useful in that it is*
* simple, makes clear some
__________
1991 Mathematics Subject Classification. Primary: 57R40; Secondary: 55T35, 57*
*Q45.
1
2 DEV P. SINHA
relevant geometry (that of the evaluation map), and is closely related to point*
* of view of Bott and Taubes
[4] which produces knot invariants in dimension three and de Rham cohomology cl*
*asses of knot spaces
more generally [7]. The second model, which is cosimplicial, makes clear analog*
*ies with loop spaces and
is especially convenient for calculations of homotopy and cohomology groups. Bo*
*th of these models make
use of compactifications of configuration spaces, essentially due Fulton and Ma*
*cPherson [13] and done in
the category of manifolds by Axelrod and Singer [2], but with some changes need*
*ed for the cosimplicial
model [31].
Applying the cohomology and homotopy spectral sequences for the cosimplicial *
*model we obtain spectral
sequences for both cohomology and homotopy groups of Emb(I, M)which converge wh*
*en the dimension
of the ambient manifold is greater than three. When M is Rkx I, the cohomology *
*spectral sequence is
reminiscent of those of Vassiliev [35] and Kontsevich [22, 23]. The calculus of*
* embeddings approach has a
few advantages.
oThere is a map directly from spaces of embeddings to their polynomial appro*
*ximations.
oBy work of Goodwillie and Klein [18] one may study embeddings of arbitrary *
*manifolds when the
domain has codimension three or greater, which is a much better range of di*
*mensions than up to half
the dimension, as Vassiliev and Kontsevich claim is possible with their tec*
*hniques.
oBecause one does not proceed through duality, it is possible to study homot*
*opy groups as well as
cohomology groups of embedding spaces.
oFor knots our models are close to the geometry needed to define the Bott-Ta*
*ubes and Kontsevich
integrals.
The homotopy spectral sequence we produce in this paper gives the first compu*
*tations of homotopy
groups of embedding spaces. The rows in the E1-term are similar to complexes de*
*fined by Kontsevich [23].
We study this spectral sequence in [27], giving explicit computations in low di*
*mensions.
When the dimension of M is three it is not known whether our models are equiv*
*alent to knot spaces,
but one can still pull back zero-dimensional cohomology classes (that is, knot *
*invariants) from our models.
These invariants seem to be connected in various ways with the theory of finite*
*-type invariants. We indicate
such connections as well as give directions for further work in Section 9.
1.1. Basic definitions, notation and conventions. We choose a variant of Emb(I,*
* M)so that a knot
depends only on its image. Fix a Riemannian metric on M. Note that many of our *
*constructions will
depend on this metric, but changing the metric will always result in changing t*
*hese constructions by a
homeomorphism, so we make little mention of the metric in general.
Definition 1.1.Define Emb(I, M)to be the space of injective C1 maps of constant*
* speed from I to M,
whose values and unit tangent vectors at 0 and 1 are specified by fixed unit ta*
*ngent vectors in @M.
By convention, throughout this paper an embedding of any interval containing *
*0 or 1 must send that
(those) point(s) to the designated points on the boundary of M, with the design*
*ated unit tangent vectors
at that (those) point(s).
We use spaces of ordered configurations, and modifications thereof, extensive*
*ly. There are two lines of
notation for these spaces in the literature, namely Cn(M) and F(M, n). Persuade*
*d by Bott, we choose to
use the Cn(M) notation. Note, however, that Cn(M) in this paper is C0n(M) in [4*
*] and that Cn[M] in
this paper is Cn(M) in [4]. Indeed, we warn the reader to pay close attention t*
*o the parentheses in our
notation, because for the sake of brevity of notation they account for all of t*
*he distinction between various
configuration spaces. We have attempted to make our choice of parentheses somew*
*hat intuitive: Cn(M) is
the open configuration space; Cn[M] is the Fulton-MacPherson compactification, *
*the most natural closure;
Cn<[M]> is a quotient of Cn[M]. In previous versions of this paper we had chose*
*n to follow the F(M, n)
line of notation, and in homage to Fulton and MacPherson we named this compacti*
*fication of F(M, n) by
FM(M, n).
THE TOPOLOGY OF SPACES OF KNOTS 3
We work with stratifications of spaces, in a fairly naive sense. A stratifica*
*tion of a space X is a collection
of subspaces {Xc} such that the intersection of the closures of any two strata *
*is the closure of some stratum.
We associate a poset to a stratification by saying that Xc Xdif Xcis in the cl*
*osure of Xd. If two spaces
X and Y are stratified with isomorphic associated posets, then a stratum-preser*
*ving map is one in which
the closure of Xcmaps to the closure of Ycfor every c.
Through section 5 we try to assume minimal knowledge of homotopy limits and o*
*ther constructions
from algebraic topology, but in section 6 we do assume such familiarity.
There are many spaces, categories and functors involved in defining our const*
*ructions, so it may be
helpful that we have tried to consistently follow some conventions. For the mos*
*t part we use capitalized
letters in the standard Roman font to denote spaces and categories. Functors ar*
*e in the caligraphic font.
1.2. Acknowledgments. The author is deeply indebted to Tom Goodwillie. As menti*
*oned in [17] (and
also just briefly in [4]), Goodwillie has known for some time that one should b*
*e able to use Theorem 6.5
to show that the polynomial approximations to spaces of knots are given by part*
*ial totalizations of a
cosimplicial space made from spaces homotopy equivalent to configuration spaces.
2.Heuristic understanding of the mapping space and cosimplicial models
Given a knot `:I ! M and n distinct times, one may produce n distinct points *
*in the target manifold
by evaluating the knot at those points. Because a knot has nowhere vanishing de*
*rivative (which we may
think of as having no infinitesimal self-intersections), one may in fact produc*
*e a collection of n unit tangent
vectors at these distinct points. Let Int( n) be the open n-simplex and Cn(M) b*
*e the configuration space
of n distinct ordered points in M, defined as a subspace of Mn, and let C0n(M) *
*be defined by the pull-back
square
C0n(M)----!(STM)n
?? ?
y ?y
Cn(M)----! Mn,
where STM is the unit tangent bundle of M.
We now define the evaluation map as follows.
Definition 2.1.Given a knot ` let evn(`):Int( n) ! C0n(M) be defined by
(evn(`))(t1, . .t.n) = (u(`0(t1)), . .,.u(`0(tn))),
where u(v) is the unit tangent vector in the direction of the tangent vector v.*
* Let evn:Emb (I, M)!
Maps(Int( n), C0n(M)) be the map which sends ` to ev`.
The evaluation map is sometimes called a Gauss map, as it generalizes the map*
* used to define the
linking number.
For any n, evn maps the embedding space injectively to Maps(Int( )n, C0n(M). *
*We cannot expect to
use evn to study the embedding space as it stands, needing to add a boundary to*
* the open n-simplex
which we may then fix in order to define a mapping space with more topology. Th*
*e appropriate boundary
turns out to not be the standard boundary of an n-simplex but the one given by *
*the Fulton-MacPherson
compactifications, which we use to compactify C0n(M) as well. These compactific*
*ations are remarkable
in that they have the same homotopy type as the open configuration spaces and t*
*hey are functorial for
embeddings. The Fulton-MacPherson compactifications are manifolds with corners*
*, and for any ` the
evaluation map evn(`) respects the stratification. Our mapping space models are*
* essentially the spaces of
stratum-preserving maps (with one important additional technical condition). Th*
*e maps from Emb(I, M)
to these models are extensions of the evn.
Our cosimplicial models are closely related to these mapping space models. R*
*ecall the cosimplicial
model for the based loop space, M (see for example [25]). The nth entry of thi*
*s cosimplicial model
4 DEV P. SINHA
is given by the Cartesian product Mn-1. The coface maps are diagonal (or öd ubl*
*ing") maps, and the
codegeneracy maps are projections (or öf rgetting" maps). The map from the loop*
* space to the nth total
space of this cosimplicial space is the adjoint of the evaluation map from the *
*simplex to Mn-1, which
is the appropriate target of this evaluation map. The map from the loop space t*
*o the total space is a
homeomorphism if n 2. Correspondingly, the homotopy spectral sequence is triv*
*ial. On the other hand,
the filtration which arises on homology gives rise to the Eilenberg-Moore spect*
*ral sequence [25].
To make a cosimplicial model of an embedding space it is natural to try to re*
*place Mn by the configu-
ration space C0n(M) as the nth cosimplicial entry, as we have seen above that C*
*0n(M) is a natural target
for the evaluation map for embeddings. The codegeneracy maps can be defined as *
*for the loop space, by
öf rgettingä point in a configuration. The coface maps are problematic, as on*
*e cannot öd ubleä point
in a configuration to get a new configuration. One is tempted to add a point cl*
*ose to the point which needs
to be doubled, but in order for the composition of doubling and forgetting to b*
*e the identity, one would
need to add a point which is "infinitesimally close." The appropriate technical*
* tool needed to overcome
this difficulty is once again the Fulton-MacPherson compactification of configu*
*ration spaces. We will find
that, while one can define diagonal maps for Fulton-MacPherson compactification*
*s, these maps do not
satisfy the cosimplicial axioms. We proceed to use a technical variant of these*
* compactifications developed
in [31] so that the cosimplicial axioms are satisfied.
3.Goodwillie's cutting method
The idea of one version of Goodwillie's cutting method is to approximate the *
*space of embeddings of a
manifold M in N with by using spaces of embeddings of M - A in N for codimensio*
*n zero submanifolds
A of M. Fix a collection {Ji}, for i = 1 to n of disjoint sub-intervals of I.
*
* S
Definition 3.1.Define ES(M) for S {1, . .,.n} toSbe the space of embeddings o*
*f I - s2SJs in M
whose speed is constant on each component of I - s2SJs.
If S S0there is a restriction map from ES(M) to ES0(M). These restriction m*
*aps commute with one
another, so if for example S0= S [ {i, j} there is a commutative square
ES(M) ----! ES[i(M)
?? ?
(1) y ?y
ES[j(M)----! ES0(M).
Hence, given a knot one can produce a family of compatible elements of ES for e*
*very non-empty S.
Conversely, such a compatible family (if n > 2) determines a knot. Now consider*
* instead of a family
which is compatible on the nose, one which is compatible up to homotopy. So, fo*
*r every square as in (1)
above we consider ff 2 ES[iand fi 2 ES[jwith an isotopy between their restricti*
*ons in ES0. Moreover,
for every cube one gets by considering all subsets T such that S T S [ {i, *
*j, k}, one has an isotopy
of these isotopies, namely a map of 2 into ES[{i,j,k}whose restriction to the *
*boundary is the three
isotopies produced for the last three faces of the cube (the last three faces b*
*eing the ones with ES[{i,j,k}
as the terminal space). Such a system is more complicated than a compatible fam*
*ily, but is in fact more
manageable from an algebraic topologist's perspective, and Goodwillie shows tha*
*t spaces of families of
punctured knots which are compatible up to homotopy are in fact good substitute*
*s for spaces of knots (see
Theorem 3.7).
The way in which one formalizes these compatible families up to homotopy is t*
*hrough the language
of homotopy limits. We pause to state some basic properties of homotopy limits.*
* A reader unfamiliar
with them is encouraged to look at [5] and [15] for a more extensive treatment.*
* A reader who has seen
homotopy limits may want to skip the next subsection.
THE TOPOLOGY OF SPACES OF KNOTS 5
3.1. Homotopy limits. Limits in the category of spaces can change dramatically *
*when the maps involved
are changed by a homotopy, which means that when a space is defined as such a l*
*imit it is difficult, for
example, to recover its cohomology and homotopy groups from those of the consti*
*tuent spaces and the
induced maps between them. For example, consider the diagram
(2) X f!Z g Y.
To see an example of a limit which is not homotopy invariant, take X and Y to b*
*oth be points. This
limit can either be a point or it can be empty, depending on whether the images*
* of X and Y coincide, a
condition which (if Z itself is path-connected and is not a single point) can c*
*hange if f or g is changed
by a homotopy. On the other hand, there are familiar cases in which this limit *
*is homotopy invariant, if
say Y !gZ defines a fiber bundle, in which case the limit is the pull-back bund*
*le. Indeed, if either f or
g is a fibration, this limit is homotopy-invariant. One may think of the homoto*
*py limit in general as the
limit of the diagram one obtains by replacing spaces and maps in the original d*
*iagram by fibrations (using
path-space constructions).
We formalize as follows. A diagram of spaces is simply a functor from a small*
* category C, which we
sometimes refer to as an indexing category, to the category of spaces. Recall t*
*hat if C is a small category,
the realization of the nerve of C, denoted |C|, is a simplicial complex with a *
*vertex for every object in
the category, an edge for every morphism, a two simplex for every composition o*
*f two morphisms, and so
forth. Also, recall that if c is an object of C, the category C # c of objects *
*over c has objects which are
maps with target c and morphisms given by morphisms in C which commute with the*
*se structure maps.
Note that the nerve of C # c is contractible since it has a final object, namel*
*y c mapping to itself by the
identity morphism. A morphism g from c to d induces a map |g| from C # c to C #*
* d.
Definition 3.2.The homotopy limit of a functor F from a small category C to the*
* category of spaces is
the subspace of Y
Maps(|C # c|, F(c))
c2C
of collections of maps {fc} such that for all morphisms c ! d in C the followin*
*g square commutes
|C # c|fc----!F(c)
? ?
|g|?y X(g)?y
|C # d|fd----!F(d).
The following special case of the definition is useful. Let C be a poset with*
* a unique maximal element
m, and let F be a functor from C to based spaces in which all morphisms are inc*
*lusions of NDR pairs in
which the subspace in each pair is closed. Note that for any object c of C the *
*category C # c is naturally a
sub-category of C # m, and that C # m is itself isomorphic to C. Hence, the |C *
*# c| define a stratification of
|C # m|, and the poset associated to this stratification is C. Similarly, if al*
*l morphisms of F are inclusions
of closed subspaces then F(m) is stratified by the F(c) with associated poset i*
*somorphic to C once again.
The following proposition follows immediately from unraveling the definitions.
Proposition 3.3.Let C be a poset with a unique maximal element m, and let F be *
*a functor from C to the
category of spaces in which all morphisms are inclusions of NDR pairs in which *
*the subspace in each pair
is closed.. Then the homotopy limit of F is the space of all stratum-preserving*
* maps from |C # m| = |C|
to F(m).
Some basic properties of homotopy limit are the following (see [15] and [5]).
oHomotopy limit is a contravariant functor from the category of diagrams of *
*spaces (that is, functors
from some small category to spaces) to the category of spaces.
6 DEV P. SINHA
oThe homotopy limit is homotopy invariant in the sense that if there is a ma*
*p of diagrams indexed
by the same category F ! G in which each F(c) ! G(c) is a (weak) homotopy e*
*quivalence, then the
induced map on homotopy limits is a (weak) homotopy equivalence.
oThere is a canonical map from the limit to the homotopy limit, as the subsp*
*ace in which each fd is
constant, which is a homotopy equivalence if "enough of the maps in the dia*
*gram are fibrations."
oIf a functor F from D to C is "left cofinal" then composition with F induce*
*s a weak equivalence on
homotopy limits. Let F # c be the category whose objects are pairs (d, f) w*
*here d is an object of D
and f is a morphism from F(d) to c, and morphisms are given by morphisms g *
*in D such that F(g)
commutes with the structure maps. A functor F is left cofinal if |F # c| is*
* contractible for every
object c 2 C.
3.2. Polynomial approximations to spaces of knots. With homotopy limits in hand*
*, we may define
Goodwillie's models. In the calculus of isotopy functors [36, 16], a suitable f*
*unctor from the category of
open subsets of a manifold to the category of spaces is polynomial roughly if i*
*t is determined by its value
on open subsets which are diffeomorphic to the disjoint union of n or fewer Euc*
*lidean balls. One must
keep track of restrictions between these distinguished open sets, so the main d*
*iagrams which appear are
cubical diagrams.
Definition 3.4.Let [n] be the category of all subsets of {0, . .,.n} where morp*
*hisms are defined by inclu-
sion. Let [n]0 be the full subcategory of non-empty subsets. A cubical diagram *
*is a functor from [n] to the
category of based spaces.
Note that the nerve of [n] is an n + 1-dimensional triangulated cube whereas *
*the nerve of [n]0 consists
of n faces of that cube and is in fact isomorphic to the barycentric subdivisio*
*n of the n-simplex.
We now formalize the notion of families of punctured knots which are compatib*
*le up to homotopy.
*
* __
Definition 3.5.Let {Ji}, for i = 1 to n be a collection of sub-intervals of I. *
*Let En(M) beSthe cubical
diagram which sends S 2 [n] to the space of embeddings ES(M), which by definiti*
*on is Emb(I- s2SJs, M)
and sends the inclusion of S S0 to the restriction map. Let En(M) be the fun*
*ctor one obtains by
precomposing En(M) with the inclusion of [n]0 in [n].
__
A cubical diagram such as En(M) determines a map from the initial space in th*
*e cube, in this case
Emb (I, M), to the limit of the rest of the cube because by definition the init*
*ial space maps to all others
compatibly. For n 3 this map is a homeomorphism, since as mentioned above a f*
*amily of compatible
embeddings of I - Jifor i = 1 to n uniquely determine an embedding of I and vic*
*e-versa. Such a cube
also determines a map from the initial space in the cube to the homotopy limit *
*of the rest of the cube.
Definition 3.6.Let PnEmb(I, M)be the homotopy limit of En(M), and let ffn be th*
*e canonical map from
Emb (I, M), which is the initial space in the cube above, to PnEmb(I, M).
PnEmb(I, M)is a degree n polynomial approximation to the space of knots in th*
*e sense of the calculus of
isotopy functors [36]. The following result states that these polynomial approx*
*imations to spaces of knots
converge when the dimension of the ambient manifold is greater than three. We t*
*ake this the starting point
of our work, which focuses on understanding these models better. We recommend c*
*onsulting [18, 16, 36] to
understand the beautiful underlying theory. A good starting point is to underst*
*and weaker statements of
the following theorem (which apply in our case to when the range manifold has d*
*imension five or greater),
which essentially use only dimension counting arguments and the Blakers-Massey *
*theorem. These simple
arguments are to appear in notes from expository lectures by Goodwillie [19].
Theorem 3.7 ([18]).If the dimension of M is greater than three, then the map ff*
*n from Emb(I, M)to
PnEmb(I, M)induces isomorphisms on homology and homotopy groups through dimensi*
*on (n-1)(dimM-
3).
THE TOPOLOGY OF SPACES OF KNOTS 7
Note that [n]0is a sub-category of [n+1]0(using the standard injection of {1,*
* . .,.n} into {1, . .,.n+1}
for definiteness), and we may choose the intervals Sj for the homotopy limit de*
*fining PnEmb(I, M)to
be a subset of those defining Pn+1Emb(I, M). Thus Pn+1Emb(I, M)maps to PnEmb(I,*
* M)through a
restriction map rn such that rnOffn+1= ffn. By the above theorem, the maps ffn *
*are inducing isomorphism
on homology and homotopy groups through a range which always increases, so we d*
*educe the following.
Corollary 3.8.If the dimension of the ambient manifold is greater than three, t*
*he map from Emb(I, M)
to the inverse limit
P0Emb(I, M) P1Emb(I, M) P2Emb(I, M) . . .
given rise to by the ffn's is a weak equivalence.
A homotopy limit of a diagram essentially only depends on the spaces and maps*
* in the diagram up
to homotopy. Because the spaces ES(M) are homotopy equivalent to C0n(M) (see th*
*e proof of Proposi-
tion 5.14), we are lead to search for models equivalent to PnEmb(I, M)which inv*
*olve configuration spaces
(with tangential data). Appropriate compactifications of these configuration sp*
*aces are essential to our
construction of such models.
4.Fulton-MacPherson compactifications and categories of trees
We use two different versions of Fulton-MacPherson compactifications of confi*
*guration spaces [13] in the
setting of manifolds as first defined by [2]. In this section we recall some of*
* their properties, all of which are
proved in [31]. In outline, we define compactifications of configuration spaces*
*, with the important property
that the inclusion of the open configuration space into each of these compactif*
*ications is a homotopy
equivalence. These compactifications are naturally stratified, and the in the c*
*ase of configurations in the
interval the compactification is isomorphic to the realization of the nerve of *
*the category of strata. When
we include tangent vectors at the points in configurations there are natural di*
*agonal (or coface) maps as
well as projections between these spaces.
We start with the Fulton-MacPherson compactification of the space of ordered *
*configurations of n points
in a manifold M.
4.1. Cn[M], the basic compactification and n, the basic category of trees. The*
* definition of
these compactifications is not entirely enlightening on its own. The structure *
*is better understood through
analysis of the stratification of these spaces, which we will give below.
In our approach, the key case to define is the case of Euclidean space. Let *
*x = (x1, . .,.xn) 2
Cn(Rk). Define ffij:Cn(Rk) ! Sk-1as sending x to the unit vector in the directi*
*on of xi- xj. De-
fine fiijk:Cn(Rk) ! I as sending x to arctan(||xi- xj||=||xi- xk||).
*
* n n
Definition 4.1.Let Cn[Rk] to be the closure of Cn(Rk) included in (Rk)n x (Sk-1*
*)(2)x I(3), whereQ
the map from Cn(Rk) to this productQis the standard inclusion on the first fact*
*or, the map i the öc simplicial" variant, tangential data, and diagonal maps. We*
* now come to
our final variant of these compactifications, a new variant which is of technic*
*al importance because its
"diagonal maps" satisfy cosimplicial identities. As in the definition of Cn[M],*
* we embed M in some Rk
and restrict the maps ffij:Cn(Rk) ! Sk-1to Cn(M).
n
Definition 4.18.Define Cn<[M]> to be the closure of Cn(Rk) in Mn x (Sk-1)(2).
First note that there is a map from Cn[M] to Cn<[M]> since a point in the clo*
*sure of Cn(M) in
n n n
Mn x(Sk-1)(2)xI(3)will project onto a point in its closure in Mn x(Sk-1)(2). In*
* [31] we show that this
map is a quotient map. We first exhibit Cn<[M]> as a union of manifolds and the*
*n sketch a description of
this quotient map.
First, to label the strata of Cn<[M]> we must enlarge the category n.
Definition 4.19.Define n to be the category of f-trees whose vertices are each*
* colored, say red or blue,
with both the root vertex and any trivalent vertex being colored blue. We defin*
*e a contraction of a set of
edges of colored f-trees as we did for f-trees with the additional requirement *
*that the initial vertex of each
contracted edge must be relabeled blue. There is a morphism in n from T to T0 *
*if T0 is obtained from T
by contracting a collection of edges and recoloring some red vertices blue.
The red vertices will correspond to spaces of collinear infinitesimal configu*
*rations, and the blue vertices
will correspond to the non-collinear infinitesimal configurations.
Definition 4.20.For n > 2 define Ln(M) to be the space whose points are lines t*
*hrough the origin in
the tangent bundle of M along with n labelled points points on each line up to *
*orientation-preserving
diffeomorphism in the line.
Thus, Ln(M) is diffeomorphic to n!=2 copies of STM.
Definition 4.21.Let ICon(M) be the subspace of ICn(M) of n points in a tangent *
*space of M which are
not all collinear up to translation and scaling.
Definition 4.22.Let T be a colored f-tree. Given a root edge e of T define ICe(*
*M) to be subspace of the
product Y Y
ICo#v(M) x L#v(M)
v2V (e),v blue v2V (e),v red
of tuples of infinitesimal configurations and lines in the tangent bundle all s*
*itting over the same point in
M, and moreover such that if v and w are red vertices which are connected by an*
* edge, then the lines
THE TOPOLOGY OF SPACES OF KNOTS 13
corresponding to v and w must be distinct. Let pe be the projection from ICe(M)*
* onto M. Define the
space CT<[M]> as a pullback as follows
CT<[M]> ----!Cr(T)(M)
?? ?
y ?y
Q Qpe r(T)
e2E(T)ICe(M)----! M .
Theorem 4.23.If M has dimension greater than one, Cn<[M]> is a union of the man*
*ifolds CT<[M]> for
T 2 n. Moreover, these manifolds define a stratification whose associated cate*
*gory is isomorphic to n.
It is not immediate from the structure given by this stratification to realiz*
*e Cn<[M]> as a quotient of
Cn[M]. We sketch briefly how to define this quotient map explicitly. First one *
*must refine the stratification
of Cn[M], by considering ICn(M) for n > 2 as the union of ICon(M) and its compl*
*ement, which is
diffeomorphic to Ln(M) x Int( n-2). One then notes that the closure of each Ln(*
*M) x Int( n-2) in
Cn[M] is diffeomorphic to Ln(M) x An-2. The quotient map from Cn[M] to Cn<[M]> *
*takes each copy of
Ln(M) x An-2and projects it onto a copy of Ln(M).
We go further in [31] to show the following.
Theorem 4.24.The projection from Cn[M] to Cn<[M]> is a homotopy equivalence. Mo*
*reover, the re-
striction of this projection to any stratum of Cn[M] is a homotopy equivalence.
As mentioned in section 2, we need versions of our configuration spaces which*
* include tangent vectors
at all of the points in a configuration.
n
Definition 4.25.Let C0n<[M]> be the closure of C0n(M) in (STM)n x (Sk-1)(2)wher*
*e the maps onto
Sk-1are defined as the composite of the projection of C0n(M) onto Cn(M) and the*
* maps ffij. Similarly,
n n
let C0n[M] be the closure of C0n(M) in (STM)nx (Sk-1)(2)x I(3).
It is easy to show that C0n[M] fits in a pull-back square similar to that whi*
*ch defines C0n(M). We choose
stratifications of these spaces so that the associated categories are still n *
*and n. Namely, to a tree T in
n we associate the pullback C0T[M] defined as
C0T[M]----!(STM)r(T)
?? ?
y ?y
CT[M]--f--! Mr(T),
where r(T) is the valence of the root vertex and f is the canonical projection.*
* There is a similar construction
for C0R<[M]>. Note that the union of the subspaces in each of these stratificat*
*ions is no longer the entire
space as it was for Cn[M] and Cn<[M]>.
Define Cn<[M, @]> as before, and for a manifold with two distinguished tangen*
*t vectors on its bound-
ary, as in the definition of Emb(I, M), define C0n<[M, @]>, respectively C0n[M,*
* @], to be the subspace of
C0n+2<[M, @]>, respectively C0n+2[M, @], which sits over points in (STM)n+2whos*
*e first and last coordi-
nates are given by the distinguished tangent vectors.
Perhaps the most commonly used maps between products Mn are diagonal maps and*
* projections, used
for example to define the cosimplicial model of the loop space of M. Unlike the*
* open configuration spaces,
these compactifications have canonical diagonal maps as well as projections. Mo*
*reover, for the Cn<[M]>
these maps satisfy cosimplicial identities.
14 DEV P. SINHA
First consider the square
Q Q
C0n(M)----! (STM)nx 1 i to
C0n-1<[M]>, which extend the lth projection from C0n(M) to C0n-1(M) and lift th*
*e lth projection map from
(STM)n to (STM)n-1.
The diagonal or coface maps are slightly more difficult to define since there*
* are no corresponding maps
on the open configuration spaces which one can extend. For purposes of reindexi*
*ng, let oel(i) equal i if
i l or i-1 if i > l. Let F be our embedding of M in Rk and let sF0be the comp*
*osite STM dF!Tx Rk!u
Sk-1, where Tx Rk is the bundle of non-zero tangent vectors of Rk and u sends a*
* non-zero vector to its
corresponding unit vector.
Definition 4.27.Let
Y Y
ffil:(STM)nx Sk-1! (STM)n+1x Sk-1
1 i by restricting ffil. We*
* prove the following in [31].
Theorem 4.28.The map ffilrestricted to C0n<[M]> maps to C0n+1<[M]>. It lifts th*
*e lth diagonal map from
(STM)n to (STM)n+1and extends the map from C0n(M) to IC2(M) xM C0n(M) C0n+1<[*
*M]> which
sends v1, . .,.vn to vi, (v1, . .,.vn).
Note that there is a version of ffilon Cn<[M]> itself once one has a non-zero*
* section of the tangent bundle
with which one may define a0i,i+1, the relative vector at the doubled point.
Both ffiland slrestrict to maps on C0n<[M, @]>, where for ffilwe have 0 l *
* n + 1 and for slwe require
0 < l < n+1. Informally, ffi0and ffin+1add a point to a configurationQat one of*
* the chosen boundary points.
It is easy to check that on the ambient spaces (STM)nx 1 i, the maps ffiiand sisatisfy the cosimplici*
*al identities.
*
* n
We now define versions of diagonal maps for C0n[M]. Let ffil(m):(STM)nx (Sk-1*
*)(2)! (STM)n+m x
n+m
(Sk-1)( 2) denote the composite of ffil with itself m times. Next, note that th*
*ere is a map from the
m+1
m - 1st Stasheff polytope Am-1 to I( 3), which we call ' given by the product o*
*f the maps fiijkin the
identification of Am-1 as Cm-1[I, @]. Also, let oemlbe defined as
8
>*l ifl i l + m
:i - m ifi > l + m.
THE TOPOLOGY OF SPACES OF KNOTS 15
n n+m- m
Finally, let øml:I(3)! I( 3 )(3)send (bijk) for 1 i < j < k n to (b0ijk) fo*
*r 1 i < j < k m + n
but not l + 1 i < j < k l + m where b0ijk= boeml(i),oeml(j),oeml(k).
Theorem 4.30.The maps
i nj n
(3) ffil(m) x øm x ': (STM)nx (Sk-1)(2)x I(3)x Am-1 !
i n+mj n+m m+1 m+1
(STM)n+m x (Sk-1)( 2)x I( 3 )-( 3)x I( 3)
restrict to maps ffil(m):C0n[M] x Am-1 ! C0n+m[M] which lift the diagonal maps *
*(STM)n ! (STM)m.
Clearly, the maps ffiiand sicommute with the quotient maps from C0n[M] to C0n*
*<[M]>.
Note that the restriction of ffil(m) to C0n[M] x v, where v is a vertex of Am*
*-1 is a composite of ffii(1)'s.
Thus, if we think of ffil(1) as trying to be a coface map, the map ffil(m) prov*
*ides canonical homotopies,
parameterized by Am-1, between composites of such coface maps which would agree*
* in a cosimplicial
setting.
5.The mapping space model
As we have mentioned, the evaluation map of a knot `, namely evn(`) from An (*
*one component of
Cn[I, @]) to C0n[M, @] is stratum preserving. But note as well that for a point*
* on the boundary of An which
has a triple point (or greater) the image of that triple point in C0n[M, @] wil*
*l be an infinitesimal triangle
(or n-gon) which is degenerate in the sense that all points are aligned along a*
* single vector, namely the
tangent vector to the knot.
Definition 5.1.A point x 2 Cn[M] is aligned if, naming its image in Mn by x1, .*
* .,.xn, for each collection
xi= xj = xk = . .,.the relative vectors vijwith i < j are all equal. A point in*
* C0n[M] is aligned if its
projection onto Cn[M] is aligned and, in the notation above, the relative vecto*
*r vijserves as the tangent
vector at xiand xj.
The subspace of a stratum consisting of points which are aligned is called th*
*e aligned sub-stratum of
that stratum.
Definition 5.2.A stratum-preserving map from An to C0n[M, @] is aligned if poin*
*ts in its image are
aligned. Let AMn(M) denote the space of aligned stratum-preserving maps from An*
* to C0n[M, @].
AMn(M) maps to AMn-1(M) by restricting an aligned map to a principal face. Le*
*t AM1 (M) denote
the homotopy inverse limit of the AMn(M). As noted above, the evaluation map ev*
*n maps Emb(I, M)
to AMn(M). The evn give rise to a map ev1 from Emb(I, M)to AM1 (M). The followi*
*ng are the main
theorems of this section.
Theorem 5.3.Let the dimension of M be greater than three. The map ev1 :Emb(I, M*
*)! AM1 (M) is
a weak homotopy equivalence.
We will relate these mapping space models to models produced by the calculus *
*of embeddings. Recall
that PnEmb(I, M)denotes the nth polynomial approximation to the space of knots *
*in M.
Theorem 5.4.AMn(M) is weakly homotopy equivalent to PnEmb(I, M)for all n includ*
*ing n = 1.
Moreover, the evaluation map evn coincides with the map ffn in the homotopy cat*
*egory.
Theorem 5.3 follows from combining Theorem 5.4 and Corollary 3.8, so the rest*
* of this section is devoted
to proving Theorem Theorem 5.4. We will first identify the mapping space model *
*as a homotopy limit.
Then, we will find a sequence of equivalences between this homotopy limit and t*
*he homotopy limit defining
PnEmb(I, M). This sequence first involves changing the shape of the homotopy li*
*mit involved and then
interpolating between configuration spaces and embedding spaces by defining a s*
*pace which incorporates
both configurations and embeddings.
16 DEV P. SINHA
For reference, we list the sequence of equivalences and the lemmas in which t*
*hey are proved now, even
though the intermediate homotopy limits have not been defined. In Lemma 5.6 we *
*show that AMn(M) is
homeomorphic to the homotopy limit of a diagram Dn[M]. We then relate the diagr*
*am Dn[M] to others
which interpolate between it and En(M), whose homotopy limit is PnEmb(I, M), as*
* outlined here.
(4) Dn[M] !5.8fDn<[M]> 5.11Dn<[M]> !5.18En<[M]> 5.18En(M).
We proceed with our sequence of lemmas.QFirst note thatPthe closure of the al*
*igned substrata of C0n[M, @]
are homeomorphic to C0n-m[M, @] x Aki, for some kiwith ki= m - 2. In fact, w*
*e see that these
strata are precisely in the image of the maps ffil(m).
Given a root edge e of a tree T, let vebeQthe terminal vertex of e and let #e*
* be the valence of veminus
three. Recall that AnrTis isomorphic to v6=vrA|v|-3, where v runs over all non*
*-root vertices of T.
Definition 5.5.Let e be an edge of T and let T0 be the tree resulting from the *
*contraction of T, so
AT = A#exAT0. Let Dn[M] be the functor from n to spaces which sends a tree T t*
*o C0|vr|-2[M, @]xAnrT,
which sends the contraction of a root edge e of T to
i j
ffii(#e + 1) x id: C0|vr|-2[M, @] xxA#eAT0! C0|vr|+#e-1[M, @] x AT0,
and which sends the contraction of a non-root edge to id x iT,T0.
The following is now straightforward, following from Proposition 3.3, Proposi*
*tions 4.13 and 4.15 and
the identification of the aligned strata of C0n[M, @] as the images of the maps*
* ffii(m).
Lemma 5.6.AMn(M) is homeomorphic to the homotopy limit of Dn[M].
Now that we have identified AMn(M) as the homotopy limit of Dn[M], we interpo*
*late between Dn[M]
and En(M). The spaces C0n<[M]> play an important role in this interpolation. *
*An aligned stratum-
preserving map from An to C0n[M] can be composed with the quotient map from C0n*
*[M] to C0n<[M]>.
Because as stated in Theorem 4.24 the restriction of the quotient map from C0n[*
*M] to C0n<[M]> to any
stratum is a homotopy equivalence, the corresponding mapping spaces are homotop*
*y equivalent. We
phrase this equivalence in terms of homotopy limits as follows.
Note that the aligned strata for C0n<[M, @]> are simply of the form C0n-k+1<[*
*M, @]>.
Definition 5.7.Let eDn<[M]> be the functor from n to spaces which sends a tree*
* T to C0|vr|-2<[M, @]>,
which sends the contraction of the ith root edge to ffii(#e), and which sends t*
*he contraction of a non-root
edge to the identity map.
There is a map of diagrams between Dn[M] and eDn<[M]> defined by quotient map*
*s. Because these
quotient maps are homotopy equivalences, we deduce the following.
Lemma 5.8.The map from the homotopy limit of Dn[M] to the homotopy limit of eDn*
*<[M]> defined by
the quotient maps is a homotopy equivalence.
For the next step in our series of equivalences, note that since many of the *
*maps in the definition of
Den<[M]> are the identity they may be eliminated. This elimination corresponds *
*to a replacement of the
category n by [n]0.
Definition 5.9.Let Dn<[M]> be the functor from [n]0 to spaces which sends S to *
*C0#S<[M, @]> and sends
the inclusion S S0where S0= S [ i to ffii(k) where k is the number of element*
*s of S less than i.
By construction, we have the following.
Lemma 5.10.Den<[M]> is the composite of Dn<[M]> with the functor Fn of Definiti*
*on 4.16.
THE TOPOLOGY OF SPACES OF KNOTS 17
The following lemma, which gives the next link in our chain of equivalences, *
*now follows immediately
from Proposition 4.17 and the fact that cofinal functors induce equivalences on*
* homotopy limits.
Lemma 5.11.The homotopy limit of eDn<[M]> is weakly homotopy equivalent to the *
*homotopy limit of
Dn<[M]>.
To interpolate between Dn<[M]> and En(M), we incorporate both embeddings and *
*configurations in one
space. As in Definition 3.5, letS{Ji}, for i = 1 to n be a collection ofSsub-in*
*tervals of I. Let {Iff} be the set
of connected components of I - s2SJs, and given an embeddingQf of I - s2SJsle*
*t fffbe the restriction
of f to Iff. Let evS be the evaluation map from ES(M) x Iffto C0#S+1(M).
Definition 5.12.Given a metric space X define H(X) to be the space whose points*
* are compact subspaces
of X and with a metric defined as follows. Let A and B be compact subspaces of *
*X and let x be a point in
X. Define d(x, A) to be lim infa2Ad(x, a). We define d(A, B) to be the greater *
*of lim supb2Bd(b, A) and
lim supa2Ad(a, B), which is sometimes called the Hausdorff metric.
Note that since STM is metrizable, as is of course Sk, then so is C0n<[M]>.
Definition 5.13.Let S 2 [n]0. Define ES<[M]> to be the union of ES(M) and C0#S*
*-1<[M, @]> as a
subspace of H(C0#S+1<[M]>), where C0#S-1<[M, @]> C0#S+1<[M]> is a subspaceQof*
* one-point subsets and
the image of f 2 ES(M) in H(C0#S+1<[M]>) is given by the image of evS(f x Iff).
It is helpful to think of ES<[M]> as a space of embeddings which may be degen*
*erate by having all of
the embeddings of components "shrinkü ntil they are tangent vectors, which we *
*think of as infinitesimal
embeddings.
Proposition 5.14.The inclusions of C0#S-1<[M, @]> and ES(M) into ES<[M]> are we*
*ak homotopy equiv-
alences.
In fact we first define a map from ES<[M]> onto C0#S-1<[M, @]> which is a def*
*ormation retraction.
Definition 5.15.Let ffl:ES<[M]> ! C0#S-1<[M, @]> be the mapQwhich is the identi*
*ty on C0#S<[M, @]> and
on ES(M) is defined by sending f to evS(f, m), where m 2 Iffhas first coordina*
*te m0= 0, last coordinate
m1= 1, and other coordinates given by defining mffto be the mid-point of Iff.
Proof of PropositionW5.14.e define a homotopy between ffl and the identity map *
*on ES<[M]> which is the
identity on C0#S-1<[M, @]>. On ES(M), letting s be the homotopy variable we set*
* fff(t)s= f((1-s)t+smff)
for s < 1.
Next, note that the restriction of ffl:ES(M) ! C0#S+1(M, @) is a fibration, e*
*ssentially bySthe isotopy
extension theorem. We show that the fiber of this map, namely the space of embe*
*ddings of I- s2SJswith
a given tangent vectors at the mff, is weakly contractible. Suppose we have a f*
*amily of such embeddings
parameterized by a compact space. First, we may apply a reparametrizing homotop*
*y (as above, of the
form f(t, s) = f((1 - s)t + smff) for s 2 [0, a] for some a) until the image of*
* each component lies in a fixed
Euclidean chart in M about the image of mff. By compactness there is an a which*
* works for the entire
family.
Noting that the unit tangent vector at mffis by definition fixed for all poin*
*ts in a fiber of ffl so we may a
priori choose coordinates in each chart around these points so that each fff(t)*
* = (fff,1(t), fff,2(t), . .).with
each fff,j(mff) = 0 and f0ff(mff) = (a, 0, 0, . .).for some a > 0. Next, for ea*
*ch component, consider the
"projection" homotopy defined in coordinates by
fff(t)s= (fff,1(t), sfff,2(t), sfff,3(t), . .)..
This homotopy is not necessarily an isotopy, but it will always be on some neig*
*hborhood of mffsince the
derivative there is bounded away from zero throughout the homotopy. By compactn*
*ess of the parameter
space, there is some non-zero b such that this homotopy is an isotopy on a neig*
*hborhood N of mffof
18 DEV P. SINHA
length b for all points in the parameter space. The composite of the first repa*
*rametrizing homotopy, a
second reparametrizing homotopy which changes the image of each interval so as *
*to be the image of N,
and the projection homotopy defines a homotopy between the given family of embe*
*ddings and one which _
is essentially a constant family (up to scaling on the first coordinates of the*
* fixed charts). |_|
Hence, the spaces ES<[M]> are a suitable interpolation between ES(M) and C0#S*
*-1<[M, @]>. Of course,
to define a suitable diagram interpolating between Dn<[M]> and En(M) we need ma*
*ps as well as spaces.
Proposition 5.16.Let S S0in [n]0where S0= S[i. There is a map æS,S0:ES<[M]> !*
* ES0<[M]> whose
restriction to ES(M) is the map to ES0(M) defined by restriction of embeddings *
*and whose restriction to
C0#S-1<[M, @]> is the map ffii.
Proof.We simply need to check that the function so defined is continuous. It is*
* continuous when restricted
to either ES(M) or C0#S-1<[M, @]>. Because ES(M) is open and dense in ES<[M]>, *
*it suffices to check
continuity on C0#S-1<[M, @]>. Since ffiiis continuous, it suffices to show that*
* for every ffl there is a ffi such
that if distance between ` 2 ES(M) and x 2 C0#S-1<[M, @]> is less than ffi then*
* the distance between
æS,S0(`) and ffii(x) is less than ffl.
*
* n
We consider C0#S-1<[M, @]> as it is defined, as a subspace of (STM)n x (Sk-1)*
*(2)where n = #S + 1,
and use the notation of Definition 4.27 so that x = ((x1, . .,.xn), (a12, . .,.*
*an-1,n)). Also, let F be the
embedding of M in Rk used to define the maps to Sk-1. A bound on the distance b*
*etween x and ` is
equivalent to a bound for each ff between xffand `0(t) for all t 2 Iff, as well*
* as a bound on the distance
between aijand the unit vector in the direction of F O `(t) - f O `(s) for t 2 *
*Iiand s 2 Ij. Such bounds
clearly give rise to the same bounds on the distance between x0fi= x0oei(fi)and*
* `0(t) for all t 2 Ifi, as well as
better bounds on all factors of Sk-1other than the factor labelled by i, i + 1.*
* In this last case, note that
we may choose our bound on the distance between xiand `0(t) for all t 2 Iiso th*
*at the distance between
sF0(xi) and s(F O `(ti) - F O `(ti+1)) for all ti2 Ii, ti+12 Ii+1is arbitrarily*
*_small.
|_|
Definition 5.17.Let En<[M]> be the functor from [n]0 to spaces which sends S to*
* ES<[M]> and sends the
inclusion S S0where S0= S [ i to æS,S0.
We have constructed the maps æS,S0so that both Dn<[M]> and En(M) map to En<[M*
*]> through the
inclusions entry-wise. By Proposition 5.14 these inclusions are weak equivalenc*
*es, so we may deduce the
following.
Lemma 5.18.The homotopy limits of Dn<[M]> and En(M) are weakly equivalent to th*
*e homotopy limit
of En<[M]>.
We may now piece together the proof of the main theorem of this section.
Proof of Theorem 5.4.The fact that AMn(M) is weakly equivalent to PnEmb(I, M)fo*
*llows from the
string of equivalences given by Lemmas 5.6, 5.8, 5.11, and 5.18.
It remains to show that the evaluation map evn coincides in the homotopy cate*
*gory with the map ffn.
Clearly evn coincides with other evaluation maps (which by abuse we also call e*
*vn) in the equivalences of
Lemmas 5.8 and 5.11. Thus we focus on the equivalences of Lemma 5.18 and show t*
*hat the composite
of evn and the inclusion of holimDn<[M]> in holimEn<[M]> is homotopic to the co*
*mposite of ffn and the
inclusion of holimEn(M) in holimEn<[M]>. Q
Recall that holimEn<[M]> is the subspace of the product S2[n]0Maps( #S-1, ES*
*<[M]>) of maps {fS}
which are compatible in that if S S0then the restriction of fS0to the face of*
* #S0-1identified with
#S-1 is the composite of fS and the restriction map from ES<[M]> to ES0<[M]>. *
*The map ffn sends
Emb (I, M)to this space as the subspace in which each fS is constant as a funct*
*ion on #S-1, with image
THE TOPOLOGY OF SPACES OF KNOTS 19
given by the restriction from Emb(I, M)to ES(M). We will homotop these fS by "s*
*hrinking towards the
evaluation points".
Let æ(J, t, s) be the interval which linearly interpolates, with parameter s,*
* between the interval J and
the degenerate interval [t, t]. Explicitly, if J = [a, b] then æ(J, t, s) is th*
*e interval [(1-s)a+st, (1-s)b+st].
Let x = 0 t1 . . .t#S-1 be a point in #S-1 and by convention let t0 = 0 *
*and t#S = 1, and
letS` 2 Emb(I, M). We define our homotopy of fS by defining hS(`, x,Ss) for sS<*
* 1 to be the embedding
of ffIffwhich is theScomposite of the linear isomorphism between ffIffand ff*
*æ(Iff, tff, s) and the
restriction of ` to ffæ(Iff, tff, s). We define hS(`, x, 1) to be (evn(`))(x).*
* It is straightforward to_check
that hS is well-defined and continuous, and that the various hS for differing S*
* are compatible. |_|
Note that Goodwillie's Theorem 3.7, upon which we build, can be proved for kn*
*ots in manifolds of di-
mension five or greater by dimension-counting arguments (sharper versions of th*
*is theorem require surgery
theory and the results of Goodwillie's thesis [14]). We wonder if one can prove*
* that the inclusion of the
embedding space into the space of aligned maps through the evaluation map can b*
*e shown to be highly
connected by more direct arguments. Is there a dimension-counting argument to d*
*etermine the greatest
number of parameters with which one can homotop an aligned map into an evaluati*
*on map of some knot?
This could be an important technical question, especially in the application of*
* these ideas to classical
knots, for which the analogue of Theorem 3.7 is not known.
6.The cosimplicial model
We now produce cosimplicial models of spaces of knots. We take as our startin*
*g point the model defined
by the homotopy limit of Dn<[M]>. We show that Dn<[M]> is a special kind of dia*
*gram indexed by [n]0,
namely it is pulled back from a cosimplicial diagram.
To set notation, we recall some standard constructions. Let be the cosimpl*
*icial category, whose
objects are the sets n_= {0, . .,.n} and where a morphism from m_to n_is an ord*
*er preserving map.
Special order preserving maps generate this category, namely siwhich is the sur*
*jection of n_onto n_-_1
which sends both i and i + 1 2 n_to i 2 n_-_1and diwhich is the inclusion of n_*
*into n_+_1for which i is
not in the image. Let n be the full subcategory of whose objects are the set*
*s i_for 0 i n.
A cosimplicial space is a functor from to the category of spaces. There is *
*a canonical example, o,
whose nth entry is n and whose coface and codegeneracy maps are inclusions of *
*faces and projections
between simplices. Given a cosimplicial space Xo let inXo be the restriction of*
* Xo to n.
Recall Theorem 4.29, that the diagonal and projection maps between C0n<[M, @]*
*> satisfy cosimplicial
axioms.
Definition 6.1.Let Co<[M]> be the cosimplicial space whose nth entry is C0n<[M,*
* @]>, whose coface maps
maps are given by the ffii, and whose codegeneracy maps are given by projection*
*s.
The following theorem is the main theorem of this section. Recall that the to*
*talization of a cosimplicial
space Xo is the space of maps from o to Xo.
Theorem 6.2.The space Emb(I, M)is weakly equivalent to the totalization of a fi*
*brant replacement of
the cosimplicial space Co<[M]>.
This theorem will follow from Proposition 6.4, which relates inCo<[M]> to our*
* models Dn<[M]>, and
Theorem 6.5, which is a general fact about homotopy limits over [n]0 which are *
*pulled back from n.
The fact that the nerve of [n]0 is isomorphic to the barycentric subdivision *
*of an n-simplex is related
to the existence of a canonical functor from [n]0 and n.
Definition 6.3.Let Gn:[n]0 ! n be the functor which sends a subset S to the ob*
*ject in n with the
same cardinality, and which sends an inclusion S S0to the composite i_~=S S*
*0~=j_, where i_and j_are
isomorphic to S and S0respectively as ordered sets. By abuse let Gn also denote*
* the composite of Gn with
the inclusion of n in .
20 DEV P. SINHA
The first step in proving Theorem 6.2 is to relate Dn<[M]> to Co<[M]>. The fo*
*llowing proposition is
immediate from unraveling Definitions 5.9, 6.1 and 6.3.
Proposition 6.4.Dn<[M]> is the composite of inCo<[M]> and Gn.
The next step in proving Theorem 6.2 is the immediate application of a genera*
*l theorem.
Theorem 6.5.Let Xo be a cosimplicial space. The homotopy limit of GnO Xo is wea*
*kly equivalent to the
nth totalization of a fibrant replacement of Xo.
Before proving this theorem in general, it is enlightening to establish its f*
*irst case. Consider the homo-
topy limit
H = holimX0!s0X1 s1X0,
where X0 and X1 are entries of a fibrant cosimplicial space Xo with structure m*
*aps s0, s1 and ß. By
definition then, s0 s1 are sections of ß, which is a fibration. We claim that t*
*his homotopy limit is weakly
equivalent to Tot1of the given cosimplicial space. The homotopy limit H natural*
*ly fibers over X02with
fiber X1, the based loop space of X1. On the other hand, the first total space*
* fibers over X0 with fiber
equal to (fiberß), so the equivalence is not a triviality.
Considering the diagram
X0 -s0---!X1----sX0
? ? 1?
?yid ß?y id?y
X0 --id--!X0-id---X0
we see H also fibers over H0, the homotopy limit of X0!idX0 idX0, which is simp*
*ly the space of paths in
X0, which is homotopy equivalent to X0 through a deformation retraction onto th*
*e constant paths. The
fiber of this map over a constant path is homotopy equivalent to (fiberß). In *
*fact if we lift the homotopy
equivalence of H0with X0defined by shrinking a path to a constant path, we get *
*a homotopy equivalence
of H with a subspace of H which is homeomorphic to Tot1Xo. Note that the standa*
*rd filtration of the
totalization of Xo is more efficient than our first filtration of H.
We break the proof of Theorem 6.5 into two theorems.
Theorem 6.6.The functor Gn is left cofinal.
The second theorem is straight from Bousfield and Kan [5].
Theorem 6.7.The homotopy limit of inXo is weakly equivalent to the nth totaliza*
*tion of a fibrant re-
placement of Xo.
These two theorems along with Proposition 6.4 give a chain of equivalences
holimDn<[M]> ~=holimGnO inCo<[M]> ' holiminCo<[M]> ' Totn(Co<[M]>),
which establishes Theorem 6.2. We now proceed to prove Theorem 6.6.
Lemma 6.8.The simplicial complex |Gn # d_| is isomorphic to the barycentric sub*
*division of the complex
whose i simplices correspond to pairs (S, f) where S n_is of cardinality i+1 *
*and f is an order preserving
map from S to d_, and whose face structure is defined by restriction of these m*
*aps.
Proof.Note that Gn # d_is a poset since [n]0 is. Since for any S, S ~=Gn(S) as *
*ordered sets, we may
consider elements of Gn # d_to be pairs (S, f) as in the statement of the lemma*
*. If S0 S, then (S0, f0)
maps to (S, f) if and only of f restricts to f0. Therefore, the subcategory of *
*objects in Gn # d_which map
to a given (S, f) is isomorphic to [#S - 1]0, whose realization is an #S - 1-si*
*mplex. Moreover, restating
from above, the realization of the objects under (S0, f0) is a face of the obje*
*cts under_(S, f) if and only if
S0 S and f0is the restriction of f. *
* |_|
THE TOPOLOGY OF SPACES OF KNOTS 21
We also use the following straightforward result.
Lemma 6.9.Let Y be a simplicial complex which is a union of simplices which are*
* indexed by a partially-
ordered set A which has a minimal element. Suppose that each simplex oea of Y ,*
* except for the minimal,
shares at least one face with some oeb< oea, and that each such oea has a face *
*which is not shared with any
such oeb. Then Y is contractible.
Proof.We build Y inductively by adjoining simplices in an order which does not *
*violate the partial ordering
of A. By assumption, at every step we adjoin a simplex along a set of faces whi*
*ch is non-empty and not
the set of all faces, so by induction we are adjoining a contractible complex a*
*long a contractible_complex
to a contractible complex at every step. *
* |_|
We are now ready to prove Theorem 6.6.
Proof of Theorem 6.6.By definition to show Gn is left cofinal is to show |Gn # *
*d_| is contractible. By
Lemma 6.9, |Gn # d_| is homeomorphic to Yn(d), which is the n-dimensional simpl*
*icial complex whose i
simplices are labeled by pairs S, f where S n_is of cardinality i + 1 and f i*
*s an order preserving map
from S to d_and whose face structure is defined by restriction of these maps.
Because any such (S, f) admits a map to some (n_, ~f), say by letting ~f(i) =*
* f(s(i)) where s(i) is the
greatest element of S which is less than or equal to i, Yn(d) is a union of its*
* n-simplices, which are indexed
by the set A of order-preserving maps from n_to d_. We put a partial order on s*
*uch maps where f g if
f(i) g(i) for all i, so that A has unique minimal and maximal elements given *
*by f(i) = 0 for all i and
f(i) = d for all i respectively. In all cases except for the maximal element, a*
*n f in A shares a faces with a
greater g, defined by increasing a single value of f. Similarly, in all cases e*
*xcept for the minimal element, an
f shares a face with a lesser g. In the case of the maximal element, the face d*
*efined by {0, . .,.n - 1} 7! d_
is not shared by any other simplex. Hence we may apply Lemma 6.9 to Yn(d) to de*
*duce_it is contractible
and finish our proof. *
* |_|
Because the spaces C0n[M, @] are more familiar than C0n<[M, @]> and are manif*
*olds with corners, one
would want to use them to define a cosimplicial model, just as as we used them *
*for the mapping space
model. Indeed, we have defined maps ffiion C0n[M, @] which one could use in con*
*junction with projections
to try to define a cosimplicial space. Unfortunately, for these compactificatio*
*ns ffiiffiiis not equal to ffii+1ffii.
These two maps are related by a canonical homotopy, and in fact we claim that o*
*ne can define a sort of A1
cosimplicial space with the C0n[M, @] as entries. Alternatively, one can try to*
* change the category so as
to capture the identities which are satisfied. One would like to define a categ*
*ory C whose relationship to
is much like the relationship between n and [n]0, and then find an appropriate*
* C-space model. We took
such an approach in an earlier version of this paper. Such a category C is touc*
*hed upon in [26] as well.
Because the algebra of our spectral sequences is ultimately cosimplicial, we op*
*ted for our current approach
as the most direct. Nonetheless, defining a model based on the combinatorics of*
* the associahedra might
be useful in further applications of the theory, in particular in connecting wi*
*th Bott-Taubes integrals and
in connecting with Kontsevich's models based on operads.
We end this section by solidifying the analogy between our cosimplicial model*
* for a knot space and
the cosimplicial model of a loop space. The machinery which has culminated in t*
*his cosimplicial model
for embeddings can be applied for immersions of an interval in M, namely Imm(I,*
* M), as well. Because
immersions may self-intersect globally we have thatSthe nth degree approximatio*
*n from embedding calculus
is a homotopy limit over [n]0 of spaces Imm(I - s2SIs, M) ~=(STM)#S-1. Followi*
*ng the arguments
in this paper, we get from these polynomial approximations a cosimplicial model*
* which has nth entry
(STM)n. This cosimplicial space is precisely the cosimplicial model for (STM)*
*, which is homotopy
equivalent to the space of immersions by theorems of Hirsch and Smale [32]. The*
* spectral sequence in
cohomology for this cosimplicial model is the Eilenberg-Moore spectral sequence*
*, which is thus analogous
to the spectral sequences we develop in Section 8.
22 DEV P. SINHA
7.The cohomology of ordered configurations in Euclidean space
The computations which we review now are standard (see [9, 10, 1]). To comput*
*e the cohomology ring
H*(Cn(Rk+1); A) for any ring A and k > 1 we appeal to the Leray-Serre spectral *
*sequence for the fibering
W k k+1 f k+1
nS ! Cn(R ) ! Cn-1(R ), where f forgets the last point in a configuratio*
*n. Assuming that
k > 1 the base of this fibration is simply connected, so the coefficient system*
* of the spectral sequence is
trivial (which is also true for k = 1 [9]). We find inductively that the E2-ter*
*m of this spectral sequence
is comprised of free modules concentrated in bi-degrees p, q such that both p a*
*nd q are divisible by k,
implying that the spectral sequence collapses at E2. Let pn(t) denote the Poinc*
*areQseries of Cn(Rk+1).
Inductively we deduce that pn(t) = (1 + (n - 1)tk)pn-1(t), so that pn(t) = n-1*
*i=1(1 + itk).
Determining the ring structure requires more detailed analysis. Recall that m*
*ap ffij:Cn(Rk+1) ! Sk
as sends (x1, . .,.xn) to xi-xj_|xi-xj|. Orient Sk and let ' 2 Hk(Sk) denote th*
*e dual to the fundamental class.
Let aij= ff*ij('). We have the following relations, the first two pulled back f*
*rom H*(Sk) and the last
simply restating graded-commutativity.
(5) aij2= 0
(6) aij= (-1)k+1aji
(7) aijalm= (-1)kalmaij
Now consider the special case of C3(Rk+1). The Leray-Serre spectral sequence *
*has
8
>A2, p, q = k, 0 ork, k
:0 otherwise
The product map from E0,k1 Ek,01! Ek,k1is an isomorphism. As the Leray-Serre s*
*pectral sequence
is a spectral sequence of algebras, we deduce that the products a12a23, a12a13a*
*nd a13a23span H =
H2k(C3(Rk+1)). Because H ~=A2, there must be a relation of the form
(8) c1a12a23+ c2a23a31+ c3a31a13= 0,
where at least one of the ciis non-zero. The cyclic group of order three acts o*
*n C3(Rk+1) by cyclically
permuting the points in a configuration, which gives rise to an action on the c*
*ohomology group H. Under
this action, the classes a12a23, a23a31and a31a13get cyclically permuted, so th*
*at the relation above also
holds after cyclic permutation of the coefficients. Because a12a23, a23a31and a*
*31a13span H which is of
rank two, the permutations of this relation must give redundant relations so th*
*at c1= c2= c3= 1.
For Cn(Rk+1) in general, we consider the map to C3(Rk+1) defined by forgettin*
*g all but the ith, jth
and kth points. Equation 8 pulls back to
(9) aijajk+ ajkaki+ akiaij= 0,
which we call the Jacobi identity.
Let Rn denote the A-algebra generated by classes xijin degree k for 1 i 6= *
*j n with relations given
by 5 through 7 and the Jacobi identity.
Theorem 7.1.As rings, H*(Cn(Rk+1)) ~=Rn.
Proof.Given what we have shown to this point, it suffices to show that the surj*
*ection Rn ! H*(Cn(Rk+1))
which sends xijto aijis an isomorphism by comparing ranks degree-wise. We may c*
*ompute the Poincare
series of Rn, which we call qn(t), inductively by noting that Rn is generated a*
*s an Rn-1-module by the
unit and the classes xin. Terms involving products of more than one xincan be r*
*educed to this basis since
the xinxjn= xijxin xijxjnthrough the Jacobi identity. We deduce that qn(t) = (*
*1+(n-1)tk)qn-1(t),
THE TOPOLOGY OF SPACES OF KNOTS 23
which is the same inductive relation obeyed by pn(t), the Poincare series of H**
*(Cn(Rk+1)). Because_
q1(t) = p1(t) = 1, we have qn(t) = pn(t) in general, which is what we wanted to*
* show. |_|
For our applications, we will need the cohomology ring of C0n(Rk+1)which is h*
*omotopy equivalent to
Cn(Rk+1) x (Sk)n. By the Künneth theorem, this cohomology ring is isomorphic to*
* that of Cn(Rk+1)
tensored with an exterior algebra on n generators, which we call b1 through bn.
To connect with the combinatorics of [4, 7, 35], we give a description of the*
* cohomology groups of
C0n(Rk+1) in terms of chord diagrams. Consider the free module generated by lin*
*ear chord diagrams with
vertices labeled 1, . .,.n and m edges. Note that a vertex need not be attached*
* to any edge. The edges
in the diagram are labeled (if k is odd) or oriented (if k is even). Let fWn,md*
*enote the quotient of this
module by the following relations:
oIf C has two edges connecting the same vertices, then C = 0.
oIf C and D are diagrams which differ by a change of orientation of an edge *
*(even case) or by the
transposition of labels of two edge (odd case), then C = -D.
oFix vertices i, j and k. Let T be a diagram with m + 1 edges, three of whic*
*h are ff connecting i to j,
fi connecting j to k and fl connecting k to i. Let T^ff(respectively, fi or*
* fl) denote the diagram with
m chords which include all chords in T, with the same labeling or orientati*
*on, except for ff. The
final relation in eCn,mis that T^ff+ T^fi+ T^fl= 0.
By construction, we have the following.
Proposition 7.2.fWn,m~=Hmk(Cn(Rk+1)).
This isomorphism is realized by sending ai1j1ai2j2.a.i.mjmto the diagram with*
* a chord between every
iland jl, ordered from 1 to m as given or oriented so that ilis the positive en*
*d of the chord.
We may similarly give a more combinatorial description of the cohomology grou*
*ps of C0n(Rk+1). Using
the Künneth theorem, we may realize Hnk(C0n(Rk+1)) as a module generated by cho*
*rd diagrams with n
vertices, i of which are marked, and m - i edges, with i ranging from 0 to m an*
*d relations as in the
definition of fWn,m. We call this module Wn,m.
8. The spectral sequences
In this section we give spectral sequences which converge to the homotopy and*
* cohomology groups of
Emb (I, M), when M has dimension at least four. We focus particular attention o*
*n the case of M = Rk.
Recall that the spectral sequence for the homotopy groups of a cosimplicial s*
*pace X is straightforward
to construct [5]. It is simply the spectral sequence for the tower of fibrations
Tot0X Tot1X . ...
Theorem 8.1.Let M have dimension four or greater. There is a second quadrant s*
*pectral sequence
converging to ß*(Emb (I, M)) whose E1 term is given by
"
E-p,q1= kersk* ßq(C0p<[M]>)).
The d1 differential is the restriction to this kernel of the map
p+1i=0(-1)iffii*:ßq(C0p-1<[M]>) ! ßq(C0p<[M]>).
We give more explicit computations of the homotopy spectral sequence when M i*
*s Rkx I in [27]. The
rows of this spectral sequence are reminiscent of complexes defined by Kontsevi*
*ch in [23].
We now discuss the cohomology spectral sequence. We proceed by taking the hom*
*ology spectral sequence
first studied in [25] and dualizing through the universal coefficient theorem. *
* The convergence of the
homology spectral sequence is more delicate than that of the homotopy spectral *
*sequence, arguments often
24 DEV P. SINHA
starting with the convergence of the standard Eilenberg-Moore spectral sequence*
* as input, and has been
studied in [3, 6, 29]. __
For a cosimplicial space Xo let H*(Xn), the normalized homology of Xn, be the*
* intersection of the
kernels of the codegeneracy maps si:H*(Xn) ! H*(Xn - 1). Theorem 3.2 of [6] sta*
*tes that the mod-p
homology spectral sequence of a cosimplicial_space Xo converges (strongly) when*
* three conditions are met,
namely_that Xn is simply connected, Hm (Xn) = 0 for m n, and for any given k_*
*only_finitely many
H m(Xn) with m - n = k are non-zero. The last two conditions are satisfied if H*
**(Xn) vanishes through
degree cn for some c > 1, which we call the vanishing condition. We check the v*
*anishing condition for
Co<[M]> where M is simply connected.
We first concentrate on Cn(M). Producing the same spectral sequence as Cohen*
* and Taylor [11]
obtained by different methods, Totaro [34] studies the Leray spectral sequence *
*of the inclusion of Cn(M)
in Mn. The stalks of the sheaf representing the cohomology of the fiber are pro*
*ducts of the cohomology of
Ci(Rk). Next note that the inclusion of C0n(M) in (STM)n has the same fibers. B*
*ecause the projections
siare compatible for Cn(M), (STM)n and Cn(Rk), we claim that the vanishing cond*
*ition for Co<[M]>
follows from the vanishing condition for Co<[Rk]>, which follows from Corollary*
* 8.4 below, and the vanishing
condition for the cosimplicial model of (STM), which is standard (the normaliz*
*ed homology vanishes up
to dimension (k + 1)n where k is the connectivity of STM).
We may deduce the following.
Theorem 8.2.Let M be simply connected and have dimension four or greater. There*
* is a second quadrant
spectral sequence converging to H*(Emb (I, M); Z=p) whose E1 term is given by
E-p,q1= coker (si)*:Hq(C0p-1(M); Z=p) ! Hq(C0p(M); Z=p).
The d1 differential is the passage to this cokernel of the map
(-1)i(ffii)*:Hq(C0p(M); Z=p) ! Hq(C0p-1(M); Z=p).
We now give a more combinatorial identification of this E1 term of our spectr*
*al sequence when M =
Rkx I. Because the cohomology of C0(Rkx I) is torsion free, we may combine the *
*spectral sequences
above to work integrally. Recall from Section 7 that Hnp(C0n(Rk+1)) ~=Wn,p, whe*
*re Wn,pis a module of
chord diagrams. Our first task is to identify the homomorphisms (sk)* in terms *
*of these modules.
Let øl(i) is equal to i if i l and i + 1 if i > l. For i, j 6= l, the map f*
*fij:Cn(Rk+1) ! Sk factors as
fføl(i)øl(j)O sl. Hence aij2 Hk(Cn-1(Rk+1)) maps to aøk(i)øk(j)2 Hk(Cn(Rk+1)) u*
*nder s*k. Translating
through the isomorphism of Proposition 7.2, (sl)* takes a chord diagram with n *
*- 1 vertices, relabels the
vertices according to øland adds a vertex, not attached to any edges nor marked*
*, labeled l. Hence the
sub-module generated by the images of (sl)* is the sub-module Nn,pof chord diag*
*rams in which at least
one vertex is not attached to an edge. The quotient map Wn,p! Wn,p=Nn,pis split*
*, so that Wn,p=Nn,p
is isomorphic to the submodule of Wn,pgenerated by chord diagrams in which ever*
*y vertex is attached or
marked, which we call Vn,p.
Next we identify the homomorphisms (ffi`)*. Recall that oe`(i) is equal to i *
*if i ` and i - 1 if i > `.
If (i, j) 6= (`, ` + 1) then the composite ffijO ffi`coincides with ffø`(i),ø`(*
*j), which implies that aijmaps to
aø`(i),ø`(j)under ffi*`. On the other hand, ff`,`+1O ffi`is the projection of C*
*0n(Rk+1) onto its `th factor of Sk,
so that ffi*`(a`,`+1) = b`. We extend these computations to define ffi*`on all *
*of H*(C0n(Rk+1)) using the cup
product.
These homomorphisms have a beautiful interpretation in terms of chord diagram*
*s. Let c`be the map
on Wn,pdefined simply by contracting the linear edge between the `th and ` + 1s*
*t vertices in a chord
diagram. Moreover, we mark the `th vertex in the new diagram if there was an ed*
*ge between these two in
the original diagram, and if this vertex is already marked, we set the result t*
*o zero.
THE TOPOLOGY OF SPACES OF KNOTS 25
Corollary 8.3.There is a spectral sequence converging to H*(E(I, Rk+1)) whose E*
*1 term is given by
(
E-p,q1= Vn,pforq = kn
0 otherwise.
with d1= (-1)`c`.
Note that up to regrading, the E2-term of this spectral sequence depends only*
* on the parity of k. At
first glance this spectral sequence looks similar to Vassiliev's [35] as well a*
*s to complexes of Cattaneo,
Cotta-Ramussino and Longoni [7]. We comment on potential relationships below.
We end by proving a vanishing result. Recall from Section 7 that the cohomolo*
*gy of Cn(Rk+1) vanishes
above degree (n-1)k, so that the cohomology of Cn(Rk+1)x(Sk)n vanishes above de*
*gree (2n-1)k. Note
also that the module Vn,pis zero of n < p=2, as each edge connects two vertices*
*. These two observations
establish the existence of vanishing lines in the spectral sequence above.
Corollary 8.4.The E1term of the spectral sequence of Corollary 8.3 vanishes abo*
*ve the line 2p-1 = q=k
and below the line p=2 = q=k.
9.Further work and open questions
This paper is meant to serve as a foundation for further study of spaces of k*
*nots. We see three
interrelated problems which may serve as guideposts for such further study.
oGain explicit understanding of cohomology and homotopy groups of spaces of *
*knots.
oDevelop invariant descriptions of (E2-approximations of) cohomology and hom*
*otopy groups of spaces
of knots, perhaps by connecting with theory of operads.
oDetermine the consequences of this study of spaces of knots for the problem*
* of classification of knots
in a three-manifold.
We end this paper by indicating what is known to us about each of these three*
* problems.
In [22] Kontsevich outlined a program to compute cohomology of spaces of knot*
*s in manifolds of di-
mension four or greater which is standard from the point of view of algebraic t*
*opology, namely to use a
spectral sequence as an upper bound and then produce enough classes to match th*
*at upper bound. There
is a similar program for homotopy groups, which one would only expect to be man*
*ageable rationally. The
spectral sequence in Theorem 8.1 is the first such spectral sequence computing *
*homotopy groups of knot
spaces. The spectral sequence of Theorem 8.2 is the first to address cohomology*
* of spaces of knots in a
general manifold. Even for Euclidean spaces, the spectral sequence in Corollary*
* 8.3 differs at first glance
from Vassiliev's [35], thus giving new progress on the first step of this progr*
*am.
The next step in the program, namely producing cohomology classes, has been c*
*arried out for de Rham
cohomology of spaces of knots in Euclidean spaces by Cattaneo, Cotta-Ramussino *
*and Longoni [7]. Gener-
alizing the techniques of Bott and Taubes [4], they use integrals in de Rham th*
*eory to produce complexes
whose cohomology map to the de Rham cohomology of the space of knots with gradi*
*ngs compatible with
the spectral sequence of Theorem 8.2. The situation is analogous to that of loo*
*p spaces, for which Chen's
iterated integrals [8] are used in some cases to prove collapse of an Eilenberg*
*-Moore spectral sequence.
The complexes of [7] would provide a lower bound for this cohomology if their c*
*ohomology maps in in-
jectively. It is shown in [7] that this map is injective in certain (bi-)degree*
*s by pairing the classes with
explicit homology classes. Hence, to complete the program of understanding rati*
*onal cohomology of knots
in Euclidean space it would suffice to show that the complexes of [7] are quasi*
*-isomorphic to the rows of
the spectral sequence of Corollary 8.3 and to find homology classes which pair *
*non-trivial with all of the
forms defined in [7]. Such explicit homology classes are desirable in any case *
*in order to have a complete
understanding of these computations.
26 DEV P. SINHA
Another computational area which is seemingly related is the homology of the *
*loopspace of a configu-
ration space, studied by F. Cohen and Gitler [12], in which combinatorics simil*
*ar to that of Theorem 8.2
appears.
The computational program is not as far along for homotopy groups. The author*
* along with Scannell
made first computations in [27]. We proved vanishing results and made computat*
*ions in low degrees,
showing the d1 differential to be of high rank but still giving plenty of non-z*
*ero classes, some of which
must survive to E1 of this spectral sequence. The geometry of the spherical fam*
*ilies of knots representing
these classes presents an interesting open question.
Even if collapse of our spectral sequences were known in some cases and we ha*
*d a handle on some
relevant geometry, our knowledge of these cohomology and homotopy groups would *
*not be complete. It
would be as if we knew that the bar complex gave computations of the cohomology*
* of the loop space in
some cases but we had no global understanding of the functor Tor. Our second pr*
*oblem above is to gain
such understanding, which could touch on fields far from topology.
There has been some progress on this problem. Kontsevich outlined an approach*
* to spaces of knots
and embeddings more generally based on the language of operads [23]. We can see*
* operads occur in our
approach since the spaces An and more generally Cn[Rk] form the entries of an o*
*perad equivalent to the
little disks operad. So we expect to be able to translate our mapping space mod*
*el and, if we "blow up"
the cosimplicial category as indicated at the end of Section 6, a cosimplicial *
*model into the language of
operads. Such connections are already bearing fruit. By applying a theorem of M*
*cClure and Smith from
their solution of Deligne's conjecture [24], a theorem saying that the totaliza*
*tion of a cosimplicial space
with a compatible operad structure is a two-fold loop space, we claim that the *
*space of knots in RkxI is a
two-fold loop space. The existence of this two-fold loop structure implies that*
* the homology of these knot
spaces is a Gerstenhaber algebra, which should be useful for explicit computati*
*ons. Note that Tourtchine
[33] has announced a Gerstenhaber structure on Vassiliev's spectral sequence, w*
*hich we conjecture agrees
with ours.
Finally, we remark on what is known about potential consequences of our work *
*in the case of knot
spaces of the most interest, namely knots in dimension three. There seem to be *
*many connections with
the theory of finite-type invariants. At a foundational level, the calculus of *
*embeddings approximates knot
spaces through homotopy limits or cosimplicial spaces while Vassiliev studies t*
*he Spanier-Whitehead duals
of these spaces through homotopy colimits or simplicial spaces (see [30]), so w*
*e are optimistic about the
possibility of direct relationships between these approaches.
As mentioned in the introduction, because there is a map from the space of kn*
*ots to our models, we
can pull back invariants of the set of components of our models to define knot *
*invariants. There are two
routes which one can take to do so. One route is to first pass from the spaces *
*defining our models (EJ(M)
or C0i<[M]>) to the free abelian groups on those spaces. For a cosimplicial spa*
*ce, one defines its homology
spectral sequence by passing to the free abelian group of each of its entries, *
*so invariants which one defines
in this way are enumerated by the cohomology spectral sequence of C<[M]>. In hi*
*s thesis under Goodwillie,
Volic has shown the group E2-2p,2pin this spectral sequence is isomorphic to th*
*e module of chord diagrams
modulo the four-term relations from finite-type knot theory. Volic has also sho*
*wn that the invariants one
pulls back from these models are in fact of finite type p. The open question is*
* as to whether one can pull
back all finite-type invariants from these models.
One approach to this question of factoring finite type invariants through our*
* models, at least with real
coefficients, would be to extend the definition of the differential forms by co*
*nfiguration space integrals on
Emb (I, M)of [4] to models closely related to ours. We suspect that an extensio*
*n should exist, noting that
general aligned stratum preserving maps share many properties with the evaluati*
*on map of a knot which
are used in the proofs of [4]. But there are serious technical issues. One impo*
*rtant piece which is missing
from our mapping space model is that the restrictions of an aligned map to the *
*various principal faces of
An need not coincide. One expects better results with a cosimplicial model (or *
*in particular a model based
on a category which is a blow-up of the cosimplicial category), but there one r*
*uns into the difficulty that
THE TOPOLOGY OF SPACES OF KNOTS 27
for example Co<[M]> is not fibrant. The categorical fibrant replacement ruins t*
*he geometry needed to do
de Rham theory, so one would need to find a geometric fibrant model.
Less is known about knot invariants one can define from our approach by not f*
*irst passing to free
abelian groups on configuration spaces. These invariants are the main topic of*
* study in [28], which
builds on [27] since these invariants are enumerated by the homotopy spectral s*
*equence. Finite-type
invariants are supposed to be analogs of linking number, which can be defined c*
*ombinatorially, analytically,
homologically or homotopically through the Gauss or evaluation map. An interpre*
*tation in homotopy
theory or differential topology of finite-type invariants has been missing from*
* the theory. The question
has been as to the correct point of view on the evaluation map. Our mapping sp*
*ace model seems to
be the right home for the evaluation map, so we are lead to search for invarian*
*ts of components of that
model. We have not fully established a connection with finite-type invariants, *
*but there is good evidence
for one. The number of these invariants agrees with the number of finite-type i*
*nvariants in low degrees.
The modules of these invariants are quotients of the modules Lie(n) used to def*
*ine the Lie operad, giving
a potential connection with Lie algebras and Feynman diagrams. We have also fou*
*nd first constructions of
these invariants through differential topology, which give a beautiful connecti*
*on with the linking number.
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Department of Mathematics, University of Oregon, Eugene OR and Department of *
*Mathematics, Brown
University, Providence RI
E-mail address: dps@math.brown.edu
*