LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS

F. R. COHEN^{∗} AND S. GITLER^{∗}

Abstract. The purpose of this note is to describe some relationships between the following topics: (1) higher dimensional variations of braids, (2) loop space homology, (3) Hopf algebras given by loop space homology, (4) natural groups attached to connected Hopf algebras,

(5) analogues of Artin’s (pure) braid group, (6) Alexander’s construction of knots arising from loop spaces, and (7) Vassiliev’s invariants of braids.

1. Introduction

The purpose of this article is to give higher dimensional analogues of braids as well as an analogue of ”braiding” certain ”pieces” of a manifold such as hyperplanes in Euclidean space or projective spaces in certain Lie groups . The main direction here is that one obtains a Hopf algebra via the homology of the loop space of a configuration space. These Hopf algebras then give groups which have properties that are analogous to Artin’s (pure) braid group, and arise from classical algebraic topology. This article is a survey of how some of these results fit [7,9,11,12,13,25]. A smattering of new results are included.

These Hopf algebras are special cases of a version of ”braiding” which occur in a broader context. In addition to the ”infinitesimal braid relations” to be defined below, there are additional relations depending on the underlying geometry of the manifold. The Lie algebras obtained in this way depend on certain naive features of the underlying manifold. Some examples are given below. One natural family of manifolds arises from the Lie groups

Date: October 1997.

1991 Mathematics Subject Classification. Primary: 55P35, Secondary: 14D99.

*Partially supported by the NSF.

2 F. R. COHEN^{∗} AND S. GITLER^{∗}

SU(n). The structures of the Hopf algebras, Lie algebras, and groups encountered here differ drastically in case of SU(3) than those for SU(n) with n not 3.

One of the Hopf algebras that occurs here is the universal enveloping algebra of the ”universal Yang-Baxter Lie algebra” which satisfies relations that are sometimes called the ”infinitesimal braid relations”, or the ”horizontal 4T relation” and ”framing independence” to knot theorists. This Hopf algebra is the universal enveloping algebra of a graded Lie algebra which also occurs in the study of the Vassiliev invariants of braids [3,18,20,21,27]. In particular, these algebras ”account” for all of the Vassiliev invariants of braids as described below in section 4, 5 and 9.

The Hopf algebras encountered here give rise to natural groups. Namely, attached to any Hopf algebra with involution is the group of coalgebra maps from a fixed coalgebra to the Hopf algebra. With a natural fixed choice of coalgebra given below, this group is filtered, and the associated graded is, as a set, the product of the primitive elements in the Hopf algebra. In the special case for which the Hopf algebra is the homology of the loop space of the configuration space for R^{2n}, the associated graded is a Lie algebra that is isomorphic to the Lie algebra associated to the descending central series for the pure braid group ( where these Lie algebras are tensored with the rational numbers ). There are further Lie algebras which take into account the underlying geometry of the manifold.

With a particular choice of coalgebra, the groups alluded to above are gotten by assembling the images of the Hurewicz homomorphism into a natural group which, as a set, is the product of the primitive element in a Hopf algebra. One example of these groups of coalgebra maps is given by the Malĉev completion of a free group [25,17]. These completions assemble themselves in various ”twisted” ways in the case where the Hopf algebra is the homology of the loop space of the configuration space for R^{n}.

Namely, the associated Lie algebras are isomorphic ( in characteristic zero ) to the ”universal Yang-Baxter Lie algebra”, but the groups themselves have different structures. The

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 3

point of this is that the Hopf algebras attached to these constructions ”see” the same quadratic commutator relations as the pure braid groups. The groups themselves are filtered and are isomorphic on the level of associated graded modules. The relations in these groups only agree modulo filtration, and the groups themselves are not actually isomorphic to the braid groups, but only have isomorphic associated graded Lie algebras [25].

Regarding braids as equivalence classes of motions of distinct particles in the plane through time, there is an extension given by replacing points by other ”pieces” of a manifold. There are natural ”braid-like” groups which are defined for any manifold which reflect these motions, and arise as a target of the classical Hurewicz homomorphism. These ”braid-like” groups satisfy relations given in the braid group up to ”quadratic terms” as well as ”extended Yang-Baxter relations” as described in section 3 provided that the underlying manifold enjoys additional naive geometric properties. These structures are developed in sections 5, and 6.

Constructions involving loop spaces of configuration spaces in turn inform on spaces of embeddings. For example, a classical result of J.W. Alexander gives that every ”knot type” is given by ”closing-up” a choice of braid [1]. Alexander’s construction arises as a map from the loop space of a configuration space to the space of embeddings of S^{1} in R^{3}, and fits in a wider context. That is, there is a map out of the loop space of a configuration space to a space of embeddings of a circle in a manifold M which when M = R^{3} specializes to Alexander’s map after applying π_{0}. One immediate observation in this context is that any component of the double loop space of smooth embeddings of S^{1} in R^{3} splits as a product of the double loop space of the 3-sphere and some other (possibly contractible) space. (This result gives a possibly bizarre connection between the braid groups and knots as Ω^{2}S^{3} is homotopy equivalent to the Quillen plus construction applied to the classifying space of the stabilized braid group.) A second related observation is a splitting for loop spaces of spaces of embeddings of circles in the product of a manifold M × R^{3} as given in section 7 here.

4 F. R. COHEN^{∗} AND S. GITLER^{∗}

The calculations here fit naturally into a context of ”quasi-embedding spaces” which are defined formally below. These ”quasi-embedding spaces” are given by a subspace of the continuous functions from a circle to a manifold M which satisfy the property that the functions restricted to certain subspaces of the circle are embeddings of the subspaces, but not necessarily embeddings. The subspaces on which the functions are required to be embeddings are given by all cosets of the circle determined by the subgroup of all 2^{r} − th roots of unity for all r > 0. In the case that the underlying manifold is R^{n}, these ”quasi-embedding spaces” are the inverse limit of spaces whose homology is given in terms of Hochschild homology of the universal enveloping algebras for the ”universal Yang-Baxter Lie algebras”. These spaces are the subject of section 8.

Based on work of T. Kohno, there is a comparison of the Hopf algebras encountered here to those that give the Vassiliev invariants. This is carried out in section 9.

The first author would like to thank the CRM for providing a stimulating mathematical environment, and for their generous support. Part of the work here was done at the CRM.

Table of contents:

1 Introduction

2 Braids and their generalizations to higher dimensions

3 ”Universal Extended Yang-Baxter Lie algebras”

4 Loop space homology of configuration space

5 Natural groups attached to connected Hopf algebras

6 Braid-like groups: analogues of Artin’s (pure) braid group

7 On Alexander’s construction of knots arising from loop spaces 8 Crude approximations to embedding spaces 9 Vassiliev invariants and the loop space homology of configuration spaces

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 5

2. Braids and their generalizations to higher dimensions

Recall Artin’s braid group [2,4,23]. The so-called ”pure braids” are those braids which leave the end-points unpermuted. A braid can be thought of as a path in the configuration space of k points in the plane R^{2}. A loop in this configuration space specifies a pure braid. There is a naive construction of such a path given by a smooth function from a circle to the configuration space which represents these braids and which is described explicitly below. Higher dimensional analogues are given by ”thickening” each strand, while explicit formulae are also given below.

Regard these thickenings as higher dimensional braids. These thickenings admit equivalence relations, and naturally give rise to a group which is very much like Artin’s braid group, but is not actually isomorphic to it. More precisely, these groups are filtered with the property that the associated graded is a Lie algebra. Furthermore, one feature of this Lie algebra is that, apart from a grading, it is isomorphic to the ”universal Yang-Baxter Lie algebra” (in characteristic zero, and to be defined below). By results in [15,19], that Lie algebra is precisely the Lie algebra attached to the descending central series of the pure braid group.

In addition, the idea of higher dimensional braiding extends to braidings of other geometric objects in a manifold. Namely, a braid may be thought of as the locus gotten by k distinct points in the plane that move through time. One might replace points by other subspaces of a manifold, and then consider the object obtained by motions of distinct subspaces through time. Specific examples of these subspaces are given below by choices of hyperplanes in Euclidean space or projective spaces in Lie groups.

There is a resulting Hopf algebra, Lie algebra, and group. Furthermore, these constructions are in a naive way extensions of classical constructions with additional universal relations. These constructions are addressed below and give rise to the notion of the ”extended Yang-Baxter Lie algebra relations”, or ”extended infinitesimal braid relations”[7].

6 F. R. COHEN^{∗} AND S. GITLER^{∗}

Recall that the classical configuration space of ordered k-tuples of distinct points in a manifold M is given by F (M, k) = {(m_{1}, m_{2}, ..., m_{k})|m_{j} ≠ m_{j} for i ≠ j} [11]. The higher dimensional analogues of representations for generators of the pure braid groups are induced by maps given in [6,10,7,12,13] with k ≥ i > j ≥ 1:

A_{i,j} : S^{n−1} → F (R^{n}, k).

In the case of n = 2, these elements represent generators of π_{1}(F (R^{2}, k)), the pure kstranded braid group. Adjoints of these maps are

B_{i,j} : S^{n−2} → ΩF (R^{n}, k).

A Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below. One feature of these graded Lie algebras is that, apart from gradings, they are isomorphic when n is even. These Lie algebras will also be elucidated below.

The maps A_{i,j} arise roughly by linking the diagonal in the i and j coordinates. Here fix points q_{i} in R^{n} given by q_{i} = 4i(v) where v is the canonical unit vector (1, 0, ..., 0). Regard z as a point of unit norm in R^{n}, and define

A_{i,j}(z) = (x_{1}, x_{2}, ..., x_{n})

where x_{t} = q_{t} if t is not i, and x_{i} = q_{j} + v.

In addition, J.W. Alexander gave a well-known construction of knots and links from braids [1,4]. Let Emb(S^{1}, M) denote the space of continuous embeddings of a circle in a fixed manifold M. In this note, there is a map

Θ : ΩF (M × [0, 1]^{2}, k) → Emb(S^{1}, M × R^{3})

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 7

which specializes to Alexander’s map on the level of π_{0} when M = point. This map then provides information about the space of embeddings Emb(S^{1}, M × R^{3}). There is an analogous version where one restricts to smooth embeddings and smooth loops. Some crude information concerning these spaces is given in sections 6, and 7.

3. ”Universal Extended Yang-Baxter Lie algebras”

The ”universal Yang-Baxter Lie algebras” are defined next [7,12]. Fix a graded abelian group V_{k}(n) with basis given by elements of degree n labelled by B_{i,j} for k ≥ i > j ≥ 1. Next, consider the free Lie algebra L[V_{k}(n)] generated by V_{k}(n). Let L_{k}(n) denote the quotient Lie algebra obtained from the ”graded infinitesimal braid relations” :

1. [B_{i,j}, B_{s,t}] = 0 if {i, j} ∩ {s, t} = ϕ,

2. [B_{i,j}, B_{i,t} + (−1)^{n}B_{t,j}] = 0 for 1 ≤ j < t < i ≤ k, and

3. [B_{t,j}, B_{i,j} + B_{i,t}] = 0 for 1 ≤ j < t < i ≤ k.

In the case that the B_{i,j} are of degree n = 1, the relations above are both redundant, and equivalent to the relations given by Kohno in [19,20]. In this case, it suffices to introduce the elements B_{j,i} and to use the sign convention that B_{i,j} = (−1)^{n}B_{j,i}.

The ”extended infinitesimal braid relations” are defined as follows:

1. [B_{i,j}, x_{s}] = 0 if {i, j} ∩ {s} = ϕ,

2. [B_{i,j}, x_{i} + x_{j}] = 0.

Fix a graded Lie algebra G over a commutative ring ℝ together with the k-fold direct sum of G given by G^{⊕k}. Assume that (1) L_{k}(n) is an ℝ-module ( replacing L_{k}(n) by L_{k}(n) ⊗_{ℤ} ℝ ), and (2) G is ℝ-free. Consider the coproduct in the category of graded Lie algebras ( over ℝ ) L_{k}(n) ∐ G^{⊕k}. Let

L_{k}(n) ∫ G

8 F. R. COHEN^{∗} AND S. GITLER^{∗}

denote the quotient of L_{k}(n) ∐ G^{⊕k} modulo the ”extended infinitesimal braid relations” in which the elements x_{i} denote the element a ⊕ x ⊕ b with x in G, a = 0^{⊕i−1}, and b = 0^{⊕n−i}. This construction is reminiscent of the classical wreath product construction on the level of groups.

It was pointed out in [7] that L_{k}(n) is a torsion free Lie algebra which is finitely generated in each degree as a module over the integers. Thus by the Poincare’-Birkhoff-Witt theorem, L_{k}(n) embeds in its’ universal enveloping algebra.

4. Loop space homology of configuration spaces

In this section, consider the homology of the loop space of the configuration space ΩF (M, k). It is sometimes the case that these homology groups are torsion free. If not, restrict to field coefficients. The following theorem was proven in [12], and [7].

Theorem 4.1. [12,7]

If M = R^{n} for n > 2, then the homology of the loop space ΩF (M, k) is isomorphic to the universal enveloping algebra of L_{k}(n − 2) as a Hopf algebra . Furthermore, the elements B_{i,j} are given by the Hurewicz image of the fundamental cycle for a sphere via the maps B_{i,j} defined in section 1.

With some additional hypotheses on the underlying manifold M, there is an analogous theorem with field coefficients 𝔽. A further naive idea of ”braiding of a subspace A” of M is required. Namely, a manifold M is said to be A-dominated provided the following hold:

1. There exist embeddings e_{i} : A → M, i = 1,2, with disjoint images.

2. The map e_{1} is isotopic to e_{2}.

3. The induced maps Ω(e_{i}) : Ω(A) → Ω(M) (with respect to different base-points) induce surjections in homology (with 𝔽 -coefficients).

Notice that the subspace A plays a role analogous to points in R^{2} as follows. Braids correspond to motions of points in the plane through time. Analogously, one might consider

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 9

subspaces A of a manifold and motions of disjoint copies of A through time. The homological property that the induced map Ω(A) → Ω(M) gives an epimorphism in homology provides a way to measure motions of disjoint copies of A through time. Furthermore, there is a ”braid-like” group attached to these homological measures; these are given in section 5.

Proposition 4.2. Let n > 3, and A = ΣCP^{n−1}. The manifolds SU(n) are A-dominated. Furthermore, the manifolds SU(3), and Sp(2) are not A-dominated for any A.

Proof. Notice that ΣCP^{n−1} embeds in SU(n) [J]. Since n > 3, there are disjoint and isotopic copies of ΣCP^{n−1} embedded in SU(n) by general position [18]. Furthermore, the homology of the loop space of ΣCP^{n−1} surjects to that of the loop space of SU(n).

The analogous result fails for SU(3), and Sp(2). Here notice that if there were two disjoint embeddings of ΣCP^{2} in SU(3), then the cup product of the three class with the five class is zero in the cohomology of SU(3) by duality. This contradicts that the product of the three dimensional generator with the five dimensional generator gives the top class. A similar argument applies to Sp(2).

The next theorem illustrates homological consequences of ”braiding” of spheres and subspaces of certain choices of manifolds.

Theorem 4.3. [7] Let M be a simply-connected m-dimensional manifold, m > 2, which satisfies

1. M = N − {point} for a manifold N,

2. w_{m−1}(τ(M)) = 0 where τ(M) is the tangent bundle of M if F = Z/2Z,

3. the euler class of τ(M) is zero if characteristic(𝔽) ≠ 2,

4. the homology Hopf algebra H_{∗}(ΩM; 𝔽) is a primitively generated Hopf algebra which is isomorphic to the universal enveloping algebra of the Lie algebra of primitive elements denoted P rimH_{∗}(ΩM; 𝔽), and

5. M is A-dominated.

10 F. R. COHEN^{∗} AND S. GITLER^{∗}

Then H_{∗}(ΩF (M, k); 𝔽) is isomorphic to the universal enveloping algebra of

L_{k}(n) ∫ G

where G is the Lie algebra of primitive elements, PrimH_{∗}(ΩM; 𝔽).

A similar theorem holds for manifolds with a Euclidean factor.

Theorem 4.4. [7]

Let M be a simply-connected m-dimensional manifold, m > 1, such that the Hopf algebra H_{∗}(ΩM; 𝔽) is primitively generated where PrimH_{∗}(ΩM; 𝔽) denotes the Lie algebra given by the module of primitive elements.Then H_{∗}(ΩF (R × M, k); 𝔽) is isomorphic to the universal enveloping algebra of L_{k}(m − 1) ∫ P rimH_{∗}(ΩM; 𝔽).

In case M = SU(n+k)/SU(n) and 𝔽 is given by the rational numbers, then the homology algebra H_{∗}(ΩF (M, k); 𝔽) is also described in [3]. The extensions with M = SU(3) are the most interesting.

There are analogous features of the above constructions given in work of Xicoténcatl [28] who considers configurations of orbits in a manifold with a free action of a group. Specific calculations apply to complex n-space minus the origin. He then obtains Lie algebras that support actions of cyclic groups, and which are analogous to the ”universal Yang-Baxter Lie algebras” with additional symmetries.

5. Natural groups attached to connected Hopf algebras

Let Hom^{coalgebra}(T [v], H) denote the set of coalgebra morphisms with source given by the tensor algebra over the integers with a single primitive algebra generator v in degree 1. Furthermore, the target H is a Hopf algebra with conjugation (antipode). Recall that this set is naturally a group with multiplication induced by the coproduct for the source and product for the target with inverses induced by the conjugation in H [24].

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 11

Throughout this section, assume that X is simply-connected. If H_{∗}(ΩX) is torsion free, then it is naturally a coalgebra by the Künneth theorem. If homology is taken with field coefficients 𝔽, then H_{∗}(ΩX; 𝔽) is a Hopf algebra with conjugation. Thus if 𝔽 is a field or if H_{∗}(ΩX; ℤ) is torsion free, then for R = ℤ or R = 𝔽,

Hom^{coalgebra}(T [v], H_{∗}(ΩX; ℝ))

is always a group. Of course, this is a special case of the hom-sets in a category where the target is a group object in the category. It is unclear whether it is informative to look at this construction within the context here. Thus this note will be limited to the remarks below.

Next specialize to the group of pointed homotopy classes of maps [ΩS^{2}, ΩX]. As a set, this group is isomorphic to the direct product of all of the homotopy groups of ΩX in case X is simply-connected. This last assertion follows at once from the well-known fact that the single suspension of ΩS^{2} is homotopy equivalent to the bouquet ∨_{n≥2}S^{n}.

Thus the group [ΩS^{2}, ΩX] may be regarded as reassembling all of the homotopy groups of ΩX into a single group. It will be seen below that the group [ΩS^{2}, ΩX] has additional structure that is not ”seen” by the additive structure of all the homotopy groups of X. Indeed one of the groups that appears in this context is very close to Artin’s braid group as will be seen in section 6. In addition, the fact that the group [ΩS^{2}, ΩX] gives the product of all of the homotopy groups for ΩX suggests that the choice of source has special features.

The groups [ΩS^{2}, ΩX], and Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)) are filtered groups via a decreasing filtration:

1. filtration j of [ΩS^{2}, ΩX] is given by those maps which are null-homotopic when restricted to the (j − 1)-skeleton

2. filtration j of Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)) is given by those maps which are trivial in homology in dimensions less than j.

12 F. R. COHEN^{∗} AND S. GITLER^{∗}

Denote the associated graded groups by

E_{0}^{j} = F^{j}/F^{j+1}.

Let G denote the group [ΩS^{2}, ΩX] or Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)). Consider the commutator map [−, −] : G × G → G. Further, let ϕ denote the morphism which sends a map to the induced map on homology:

ϕ : [ΩS^{2}, ΩX] → Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)).

Theorem 5.1. [9] Assume that H_{∗}(ΩX) is torsion free.

1. The function

ϕ : [ΩS^{2}, ΩX] → Hom^{coalgebra}(H_{∗}(ΩS^{2})), H_{∗}(ΩX))

is a morphism of filtered groups.

2. In case G = [ΩS^{2}, ΩX], the associated graded E_{0}^{j} is isomorphic to π_{j}(ΩX).

3. In case G = Hom^{coalgebra}(H_{∗}(ΩS^{2})), H_{∗}(ΩX)), the associated graded E_{0}^{j} is isomorphic to the module of primitives concentrated in degree j,

P rimH_{j}(ΩX).

4. On the level of associated graded groups, the map

E_{0}^{j}(ϕ) : π_{j}(ΩX) → P rimH_{j}(ΩX)

is the Hurewicz homomorphism.

5. The Hurewicz homomorphism π_{∗}(ΩX) → P rimH_{∗}(ΩX) surjects to the module of primitives in H_{∗}(ΩX) if and only if ϕ is a surjection.

6. The commutator map [−, −] : G × G → G induces a homomorphism on the level of associated groups

E_{0}^{j} ⊗ E_{0}^{k} → E_{0}^{j+k}

which endows the associated graded groups ⊕_{j≥1}E_{0}^{j} = F^{j}/F^{j+1} with the structure of graded Lie algebra ( with the usual exceptions for the primes 2 and 3 where it will be assumed for convenience that the primes 2 and 3 are units); this structure on the source is induced by the Samelson product < −, − >. Namely the Lie bracket [α, β] is given by \binom{j+k}{k} < α, β >.

7. The morphism E_{0}^{j}(ϕ) is a morphism of graded Lie algebras.

8. The associated graded module for the group

Hom^{coalgebra}(H_{∗}(ΩS^{2})), H_{∗}(ΩX))

is given by

⊕_{j≥1}E_{0}^{j} = ⊕_{j≥1}P rimH_{j}(ΩX).

As a Lie algebra the Lie bracket [a, b] is given by \binom{j+k}{k}(a ⊗ b − (−1)^{jk}b ⊗ a) for primitive elements a of degree j, and b of degree k.

Remarks: The graded module ⊕_{j≥1}E_{0}^{j} = F^{j}/F^{j+1} is not, without additional assumptions, a graded Lie algebra as [x, x], and [[x, x]x] may be non-zero. In case x is of even degree, and [x, x] is non-zero, it is of order 2. In case x is of odd degree, and [[x, x], x] is non-zero, it is of order 3. Furthermore, the binomial coefficients in part (8) above arise from the structure of the coproduct for T[v].

In addition, a theorem of [24] concerning the structure of rational homotopy groups of ΩX gives information at once about the group [ΩS^{2}, ΩX]. Indeed, this theorem in [24] concerning the rational homology of a loop space of a simply-connected space X (which is isomorphic to the universal enveloping algebra of the rational homotopy Lie algebra of ΩX ) is the associated graded version of the next theorem concerning filtered groups.

14 F. R. COHEN^{∗} AND S. GITLER^{∗}

Theorem 5.2. [9] Let X denote the rationalization of a simply-connected space of finite type. Then

ϕ : [ΩS^{2}, ΩX] → Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)).

is an isomorphism of filtered groups.

The structure of the group Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)) is frequently non-trivial as all of the homotopy groups assembles into a single group with non-trivial extensions. It is not yet clear what precisely determines these extensions as well as what the characteristic classes of the group extensions actually are. The following example gives one case where they are non-trivial and where the constructions in the above paragraph are useful.

Theorem 5.3. [24]

If X is a bouquet of q copies of the 3 − sphere, S is a set of cardinality q, and F [S]_{M} is the Malĉev completion of F[S], then there is an isomorphism of groups

F [S]_{M} → Hom^{coalgebra}(H_{∗}ΩS^{2}, H_{∗}(ΩX; ℚ)).

It is the purpose of the next section to describe some explicit examples which use the above theorem as a further connection between the braid groups and the ”Yang-Baxter Lie algebras”. Namely, the subject of the next section uses the previous theorem as well as the next theorem where the ring ℝ is assumed to be a PID.

Theorem 5.4. [9] Let

(†) : 1 → A → B → C → 1

be a short exact sequence of Hopf algebras with conjugation (antipode) that are free over ℝ, and which is split as coalgebras. Then for any fixed choice of coalgebra, there is a short exact sequence of groups (‡) : 1 ⟶ Hom^{coalgebra}(K, A) ⟶ Hom^{coalgebra}(K, B) ⟶ Hom^{coalgebra}(K, C) ⟶ 1.

Furthermore, if the short exact sequence (†) is also multiplicatively split, then the extension (‡) is multiplicatively split and the group Hom^{coalgebra}(K, B) is a semi-direct product of the groups Hom^{coalgebra}(K, A), and Hom^{coalgebra}(K, C).

The next proposition is a remark which follows from the work in [8, 24].

Proposition 5.5. Let X denote the localization at p of the (2n+1)-connected cover of the (2n+1)-sphere. Then the group [ΩS^{2}, ΩX] has exponent

1. p^{n} if p is odd, and

2. bounded above by 2^{(3n/2)+ϵ} for ϵ = 0 or 1 .

Proof. The odd primary case follows at once from

1. the fact that S^{2n+1} localized at an odd prime is an H-space. This gives that [ΩS^{2}, ΩS^{2n+1}] splits as a group [∨_{t≥2}S^{t}, ΩX], and

2. the exponents given in [8].

The 2^{q} −th power map on ΩS^{2n+1} is homotopic to the looping of the degree 2^{q} map if q > 1 after localization at 2. Thus if q > 1, the effect of the 2^{q} − th power map on [ΩS^{2}, ΩS^{2n+1}] is induced by the self-map of [ΣΩS^{2}, S^{2n+1}] given by the degree 2^{q} map on the target which is then given by the induced map [∨_{t≥2}S^{t}, ΩX] → [∨_{t≥2}S^{t}, ΩX] given by multiplication by degree 2^{q} on each homotopy group.

These bounds on exponents on homotopy groups are given by results in [26]. The proposition follows.

16 F. R. COHEN^{∗} AND S. GITLER^{∗}

6. Braid-like groups: analogues of Artin’s (pure) braid group

The main point of this section is that the group of coalgebra maps in the previous section provide groups which have features that are analogous to Artin’s (pure) braid group. Some of these ”braid-like” properties are described below. The main theme is that these groups of coalgebra maps are filtered as above. The Lie algebras attached to these filtrations via Theorems 5.1 and 5.4 are isomorphic to the Lie algebra obtained from the descending central series for the pure braid group in the cases where the underlying manifolds are given by R^{2n} for n > 1 ( and thus give the Vassiliev invariants of braids by work of Kohno [19,20] as is pointed out in section 8). There are generalizations of these ”braid-like” groups for other choices of manifolds.

Consider the groups and maps defined in the last section where X is the configuration space of ordered k-tuples of distinct points in a manifold. The first result in this direction is to consider the group Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)) as well as the filtration described above.

Theorem 6.1. If X = F (R^{n}, k) for n > 2, and homology is taken with coefficients in the rational numbers, then the associated graded Lie algebra for the group of coalgebra morphisms Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩX)) is isomorphic to the ”universal Yang-Baxter Lie algebra” L_{k}(n − 2) ⊗_{Z} Q.

The proof of this theorem follows at once from Theorems 5.3, and 5.4 above. Namely, the fibrations in [14] imply that the homology of ΩF (R^{n}, k) is given by an iterated split extension as in Theorem 5.3 where the Hopf algebra kernels at each stage are given by a primitively generated tensor algebras [7,12]. The group extensions follow by direct calculation.

Notice that apart from a grading, the Lie algebras that occur in Theorem 6.1 when n is even are isomorphic to the Lie algebra obtained from the descending central series for Artin’s braid group on k strands [15,19]. This raises the question of whether the groups themselves are isomorphic at least after forming the nilpotent completion.

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 17

This turns out to not hold. These groups of coalgebra morphisms do not have the same relations. Namely, the relations in the pure braid group can be given in terms of commutators of weight 2 and weight 3. The relations in the group of coalgebra maps satisfy these up to quadratic terms, and the groups on the level of associated graded modules agree. The actual relations in the group of coalgebra maps appear to be given by an infinite product of commutators of arbitrarily large length and is work in progress [25].

Since the relations for the pure braid group are not usually listed in terms of commutators, this presentation is recorded next. This formulation follows at once from the presentation listed in [23, p. 174] where the element [g,h] in a group G is given by the commutator ghg^{−1}h^{−1}:

I. Generators for the k-stranded pure braid group P Br_{k} are given by B_{j,i} for 1 ≤ j < i ≤ k, and

II. A complete set of relations is given by

1. [B_{r,s}, B_{i,k}] = 1 for s < i or k < r,

2. [B_{k,s}, B_{i,k}] = [B_{i,s}^{−1}, B_{i,k}] for i < k < s, and

3. [B_{r,s}, B_{i,k}] = [[B_{i,s}^{−1}, B_{i,r}^{−1}], B_{i,k}] for 1 < r < k < s.

By construction the map

ϕ : [ΩS^{2}, ΩF (R^{n}, k)] → Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}ΩF (R^{n}, k))

is a surjection. The reason for this statement is that the Hurewicz homomorphism surjects to the module of primitives in H_{∗}(ΩF (R^{n}, k)) ).

Theorem 6.2. The group [ΩS^{2}, ΩF (R^{n}, k)] is the semi-direct product of

Hom^{coalgebra}(H_{∗}ΩS^{2}, H_{∗}ΩF (R^{n}, k))

and the kernel of ϕ.

18 F. R. COHEN^{∗} AND S. GITLER^{∗}

That the infinitesimal braid relations are satisfied in the associated graded for the group of coalgebra maps Hom^{coalgebra}(H_{∗}ΩS^{2}, H_{∗}(ΩF (R^{n}, k)) suggests calling these groups ”braid-like groups”. If M is a manifold, there are similar ”braid-like groups” given by

Hom^{coalgebra}(H_{∗}ΩS^{2}, H_{∗}ΩF (M, k))

.

Theorem 6.3. If M is a simply-connected m-dimensional manifold for m > 2, then the Lie algebra given by the associated graded for the group

Hom^{coalgebra}(H_{∗}(ΩS^{2}), H_{∗}(ΩF (M × R, k); ℚ))

is isomorphic to L_{k}(m − 1) ∫ P rimH_{∗}(ΩM; ℚ).

In case M is SU(n), SO(n) or Sp(n), or certain associated homogeneous spaces, then similar results apply. The rank two cases of SU(3), and Sp(2) are more complicated than the cases of larger rank. Specific answers are given in [7].

7. On Alexander’s construction of knots arising from loop spaces

By ”closing up” pure braids, Alexander gave a procedure for constructing isotopy classes of smooth knots, and links [1,4]. This construction ”fits” with the natural maps considered here, and there are similar constructions which apply to links by using paths. There is an explicit map given below which for M = point that corresponds to ”cyclic closure” of pure braids and is induced by applying π_{0} to a geometric map between function spaces given as

Λ_{k} : ΩF (M × [0, 1]^{2}, k) → Emb(S^{1}, M × R^{3}).

Namely, an element in the loop space of the configuration space is a k-tuple of functions that do not simultaneously coincide. By (i) dividing the interval into 2k subintervals, (ii) ”stringing out” these k functions over k alternating intervals via a time parameter, and (iii) ”connecting” these strings in a compatible way on the complementary k alternating intervals, one gets an embedding of a circle. A formal description is as follows. Consider an element

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 19

in the space ΩF (M × [0, 1]^{2}, k). That is a map from the circle to F (M × [0, 1]^{2}, k) which consists of

1. a k-tuple of continuous functions (f_{1}, f_{2}, ..., f_{k}),

2. f_{i} : [0, 1] → M × [0, 1]^{2},

3. (f_{i})(0) = (f_{i})(1) = x_{i} for fixed and distinct x_{i}, and

4. (f_{i})(t) ≠ (f_{j})(t) when i ≠ j.

Next fix paths p_{i} : [0, 1] → M × R^{2} × R such that

1. (p_{i})(0) = (x_{i}, 1),

2. (p_{i})(1) = (x_{i+1}, 0) for i < k,

3. (p_{k})(1) = (x_{1}, 0),

4. the image of p_{i}((0, 1)) is contained in the complement of M × [0, 1]^{2} in M × R^{3}, and 5. the images p_{i}([0, 1]) satisfy p_{i}([0, 1]) ∩ p_{j}([0, 1]) = ϕ for i ≠ j.

The construction of the map Λ_{k} follows directly: Consider the intervals [i/2k, (i + 1)/2k], and define Λ_{k} in a piecewise manner as follows ( which admits a ”smooth analogue” ) :

1. Λ_{k}(t) = (f_{i}([2k(t − 2i/2k]), 2k[t − 2i/2k]) if 2i/2k ≤ t ≤ (2i + 1)/2k.

2. Λ_{k}(t) = (p_{i}(2k[t − (2i + 2)/2k]) if (2i + 1)/2k ≤ t ≤ (2i + 2)/2k.

A result of Alexander [1] then admits a function space analogue which follows from his results together with the definition of the maps Λ_{k}.

Proposition 7.1. In case M = point, the smooth analogue

⨿_{k≥2}{π_{0}Λ_{k}} : ⨿_{k≥2}{π_{0}Ω^{smooth}F (R^{2}, k)} → π_{0}Emb^{smooth}(S^{1}, R^{3})

induces a surjection of sets.

20 F. R. COHEN^{∗} AND S. GITLER^{∗}

Proof. Notice that the definition of the maps Λ_{k} induce the map on the level of pure braids that is given by cyclic closure of pure braids. By Alexander’s results [1], cyclic closure of pure braids gives all isotopy classes of knots for some choice of pure braid. The result follows.

Consider the natural inclusion of Emb(S^{1}, M) in the free loop space ΛM.

Proposition 7.2. If M is simply-connected, the composite

Ω(M × [0, 1]^{2}) → Emb(S^{1}, M × R^{3}) → Λ(M × R^{3})

is homotopic to the natural inclusion of Ω(M) in Λ(M). Thus the loop space of

Emb(S^{1}, M × R^{3})

is homotopy equivalent to Ω^{2}M ×X_{M} for some choice of space X_{M}, and π_{∗}(Emb(S^{1}, M ×R^{3})) contains π_{∗}(Ω(M)) as a direct summand.

Proof. The composite Ω(M × [0, 1]^{2}) → Emb(S^{1}, M × R^{3}) → Λ(M × R^{3}) is homotopic to the map which sends a loop to that loop plus the loop sum with a fixed loop. Since M is simply-connected, this map is homotopic to the natural inclusion of Ω(M) in Λ(M). The result follows from the fact that the map Ω(M) → Λ(M) is split after looping once.

Let Emb_{β}(S^{1}, R^{3}) denote the path-component of an embedding β. An analogous standard observation is that SO(3) acts on the space of embeddings Emb(S^{1}, R^{3}). Thus if β is a fixed embedding, there are maps

{β} × SO(3) → Emb_{β}(S^{1}, R^{3}) → S^{2}.

Proposition 7.3. There is a homotopy equivalence

Ω^{2}Emb_{β}(S^{1}, R^{3}) → Ω^{2}S^{3} × X_{β}

for some choice of (possibly contractible) space X_{β}.

Thus, the homotopy groups of the component of β, Emb_{β}(S^{1}, R^{3}), is non-zero in arbitrarily large degrees. An analogous result holds in the case of smooth embeddings Emb_{β}^{smooth}(S^{1}, R^{3}).

Proof. Consider the composite of the maps

{β} × SO(3) → Emb_{β}(S^{1}, R^{3}) → S^{2}

If α is an element of SO(3), this composite sends α to α(q)/||α(q)|| where q = β(1)−β(−1). A choice of rotation sending q to (1,0,0) in R^{3} gives that the above composite is homotopic to the standard map of SO(3) to S^{2} (analogous to Gauss ”linking”). Looping the map SO(3) → S^{2} twice gives a homotopy equivalence on connected components given by Ω^{2}S^{3}.

Notice that if β is smooth, then the map g takes values in Emb_{β}^{smooth}(S^{1}, R^{3}). The proposition follows.

One is led to wonder whether the spaces X_{β} have homotopy types that are independent of β. One consequence of a preprint due to A. Hatcher gives that the spaces X_{β} are contractible. A second question is whether there are function space interpretations of Markov moves for the maps given in Proposition 7.1. Proposition 7.3 gives yet another (weird ?) connection between knots and the braid groups as the Quillen plus construction for the classifying space of the stable braid group gives Ω^{2}S^{3}.

8. Crude approximations to embedding spaces

The space of continuous embeddings of a circle in a manifold admits a natural embedding as a closed subspace of the space of all (free) continuous maps of S^{1} to M, ΛM. There is a natural tower of spaces that factors this natural inclusion and is described in more detail below.

22 F. R. COHEN^{∗} AND S. GITLER^{∗}

Consider the subspace of continuous maps of a circle to the manifold which restrict to embeddings on all cosets of the group G generated by all 2^{k} − th roots of unity in a circle for k > 0. Call this subspace Emb_{G}(S^{1}, M).

This subspace is the main object of study in this section. In particular, it is pointed out below that Emb_{G}(S^{1}, M) is the inverse limit of identifiable spaces which arose in the previous sections. One consequence is that when the manifold M is Euclidean n-space for n > 2, the rational homology of each space in this inverse limit is isomorphic to the Hochschild homology of the universal enveloping algebra for the ” universal Yang-Baxter Lie algebra ” as defined above and in [7,12].

Let Γ_{k} denote the subgroup of the circle generated by the k − th root of unity ρ_{k} given by e^{2πi/k}. The inclusion of Γ_{2^{j}} in Γ_{2^{j+1}} which sends ρ_{2^{j}} to (ρ_{2^{j+1}})^{2} induces a tower of spaces

\xymatrix{X_{j+1} \ar[r] & F (M, 2^{j+2}) \ar[d]^{π_{2^{j+1}}} \\ X_{j} \ar[r] & F (M, 2^{j+1}) \ar[d]^{π_{2^{j}}} \\  & F (M, 2^{j})}

where

1. π_{2^{j}} : F (M, 2^{j+1}) → F (M, 2^{j}) is the projection induced by the inclusion of Γ_{2^{j}} in Γ_{2^{j+1}}, and

2. the fibre of π_{2^{j}} is X_{j}, the space F (M − Q(2^{j}), 2^{j+1} − 2^{j}) where Q(2^{j}) is a subset of M having cardinality 2^{j}.

Consider the inverse limit of the spaces F (M, 2^{j}), \underleftarrow{lim} F (M, 2^{j}). Notice that there is a map Θ : Emb(S^{1}, M) → Λ \underleftarrow{lim} F (M, 2^{j}) which is defined by the projection to ΛF (M, k) given by

Θ(f)(z) = (f(z), f(ρ_{k}(z)), f(ρ_{k}^{2}(z)), ..., f(ρ_{k}^{k−1}(z)).

LOOP SPACES OF CONFIGURATION SPACES, BRAID-LIKE GROUPS, AND KNOTS 23

Furthermore, each map F (M, 2^{j+1}) → F (M, 2^{j}) is the projection map in a fibre bundle and, if M = R^{n} × N, then this fibration has a cross-section [11]. Thus in the cases for which M = R^{n} × N the homotopy groups of \underleftarrow{lim} F (M, 2^{j}) are given by the inverse limit of the homotopy groups of F (M, 2^{j}) [5]. In addition, these maps have more structure some of which is described next.

Consider the subspace of ΛF (M, k) given by the Z/kZ-equivariant maps from S^{1} to F (M, k) where Z/kZ acts on S^{1} by multiplication by ρ_{k} = e^{2πi/k}. Write Λ^{Z/kZ}F (M, k) for the space of continuous Z/kZ-equivariant maps from S^{1} to F (M, k). The next remark is stated as a proposition .

Proposition 8.1. The map Θ takes values in the subspace of Λ \underleftarrow{lim} F (M, 2^{j}) given by the inverse limit of the Λ^{Z/kZ}F (M, k) for k = 2^{j}, \underleftarrow{lim} Λ^{Z/kZ}F (M, k).

The rest of this section consists of some information concerning the inverse limit of the Λ^{Z/2^{j}Z}(F (M, 2^{j})), \underleftarrow{lim} Λ^{Z/2^{j}Z}(F (M, 2^{j})), and which will be written

Λ^{Z/2^{∞}}F (M, Z/2^{∞}).

Notice that restriction to the subspace of maps which are embeddings on cosets G, Emb_{G}(S^{1}, M), is far from the space of embeddings. Namely, there are maps which have multiple points at irrational points of the circle.

Proposition 8.2. 1. There is a commutative diagram

\xymatrix{Emb(S^{1}, M) \ar[r] \ar[d]^{1} & Λ^{Z/2^{∞}}F (M, Z/2^{∞}) \ar[d] \\ Emb(S^{1}, M) \ar[r]^{inclusion} & ΛM.}

24 F. R. COHEN^{∗} AND S. GITLER^{∗}

2. The space Λ^{Z/2^{∞}}F (M, Z/2^{∞}) is a homeomorphic to the subspace of ΛM given by the continuous maps of S^{1} to M which are embeddings when restricted to cosets of G in S^{1}, Emb_{G}(S^{1}, M) .

3. If M is a simply-connected manifold of dimension at least 3, then the rational homology of Emb_{G}(S^{1}, M) is not of finite type.

Proof. Statement 1 is clear.

Consider the space Emb_{G}(S^{1}, M). Notice that the map Θ extends to give a map Θ′ : Emb_{G}(S^{1}, M) → \underleftarrow{lim} Λ^{Z/2^{j}Z}F (M, 2^{j}).

Furthermore, the projection map π : \underleftarrow{lim} Λ^{Z/2^{j}Z}F (M, 2^{j}) → ΛM takes values in the space Emb_{G}(S^{1}, M). Both natural composites π ∘ Θ′, and Θ′ ∘ π give the identity. Thus statement 2 follows.

To prove statement 3, notice that if M is R^{n} for n > 2, then the first non-vanishing homology group of \underleftarrow{lim} Λ^{Z/2^{j}Z}F (M, 2^{j}) is torsion free and not finitely generated. In case M is simply-connected, the first dimension where the homotopy is non-trivial gives that the first non-vanishing homotopy group of Emb_{G}(S^{1}, M) is not finitely generated.

Let X be a space with a free Z/kZ-action, and Λ^{Z/kZ}X the space of continuous Z/kZequivariant maps. The following is well-known.

Proposition 8.3. If X is simply-connected, the natural inclusion of Λ^{Z/kZ}X in ΛX is a rational homotopy equivalence.

Proposition 8.4. The space Λ^{Z/2^{j}Z}(F (M, 2^{j})) is rationally homotopy equivalent to ΛF (M, 2^{j}) if M is a simply-connected manifold of dimension at least 3.

There is a further interpretation of the above remarks. Define the space of almost embedded curves to be the homotopy orbit space

ESO(2) ×_{SO(2)} Emb_{G}(S^{1}, M).

Consider reduced simplicial sets X. Let ΩX denote the simplicial (Kan) loop group of X, and k[ΩX] the simplicial group algebra over a field k. Then by [16] or [22, Corollary 7.3.14],

1. the Hochschild homology of k[ΩX] is isomorphic to H_{∗}(Λ|X|; k), and

2. the cyclic homology of k[ΩX] is isomorphic to H_{∗}ESO(2) ×_{SO(2)} Λ|X|; k).

Corollary 8.5. 1. The rational homology of Λ^{Z/kZ}F (M, 2^{j}) is isomorphic to the Hochschild homology of the singular chain complex for ΩF (M, 2^{j}). If M = R^{n} for n ≥ 3, the rational homology of Λ^{Z/kZ}F (M, k) is isomorphic to the Hochschild homology of the universal enveloping algebra of the ”universal Yang-Baxter Lie algebra” L_{k}(n − 2) .

2. The space ESO(2) ×_{SO(2)} Emb_{G}(S^{1}, M) is homeomorphic to

\underleftarrow{lim} ESO(2) ×_{SO(2)} Λ^{Z/kZ}F (M, k.

The rational homology of ESO(2) ×_{SO(2)} Λ^{Z/kZ}F (R^{n}, k) is given in terms of the cyclic homology of the universal enveloping algebra for the ”universal Yang-Baxter Lie algebra” L_{k}(n − 2).

These constructions also fit in a different context one of which is discussed in the next section.

9. Vassiliev invariants and the loop space homology of configuration spaces

The purpose of this section is to describe how the homology of the loop space of a configuration space ”accounts” for all of the Vassiliev invariants of pure braids. The main additional

26 F. R. COHEN^{∗} AND S. GITLER^{∗}

input is given in work of Kohno [20]. The information here is a direct comparison of those results with the results described above.

Let V_{k}^{n} be the vector space of Vassiliev invariants of order k for pure braids with n strands ( as in [22, p. 130 ] ). Let A_{k}^{n} be the complex vector space spanned by horizontal chord diagrams with n vertical strands” modulo the ”horizontal 4T relation” and the ”framing independence relation”; these last two relations are precisely the ”infinitesimal braid relations” of section 3, and given by Kohno [22,Proposition 4.3] who proves that there is an isomorphism of complex vector spaces

V_{k}^{n}/V_{k−1}^{n} → Hom_{ℂ}(A_{k}^{n}, ℂ).

Consider the graded algebra A_{∗}^{n} which in degree k is given by the complex vector space A_{k}^{n}, and is ℂ in degree zero. Thus the Euler-Poincare’ series for A_{∗}^{n} is given by Σ_{k≥0}A_{k}^{n}t^{k} in [22,Proposition 1.3.11]

Proposition 9.1. 1. There is an isomorphism of algebras

Γ : H_{∗}(ΩF (R^{3}, n); ℂ) → A_{∗}^{n}.

2. There is an equality of Euler-Poincare’ series

χH_{∗}(ΩF (R^{3}, n); ℂ) = Σ_{k≥0}A_{k}^{n}t^{k}.

3. If q ≥ 1, then H_{∗}(ΩF (R^{q+2}, n); ℤ) is torsion free of finite type, and

χH_{∗}(ΩF (R^{q+2}, n); ℤ) = [(1 − t^{q})(1 − 2t^{q})....(1 − (n − 1)t^{q})]^{−1}, and
χH_{∗}(F (R^{q+1}, n); ℤ) = (1 + t^{q})(1 + 2t^{q})....(1 + (n − 1)t^{q}).

Proof. To prove parts (1), and (2), it suffices to give a map of algebras which induces the equality given in part (2). The algebra A_{∗}^{n} is generated as an algebra by elements B_{i,j} subject to the infinitesimal braid relations. Thus there is a surjection of algebras

Γ : H_{∗}(ΩF (R^{3}, n); ℂ) → A_{∗}^{n}.

That this map is an isomorphism follows by a comparison of their Euler-Poincare’ series given in the next calculation together with the series

Σ_{k≥0}A_{k}^{n}t^{k}

which is given by

[(1 − t)(1 − 2t)....(1 − (n − 1)t)]^{−1}

by [K2]. If q ≥ 1, the space ΩF (R^{q+2}, n) is homotopy equivalent to a product Π_{1≤i≤n−1}Ω(∨_{i}S^{q+1}) [7]. The formulas below follow at once from the Euler-Poincare series for a tensor algebra and the calculation of H_{∗}(F (R^{q+1}, n); ℤ) in [C]:

χH_{∗}(ΩF (R^{q+2}, n); ℤ) = [(1 − t^{q})(1 − 2t^{q})....(1 − (n − 1)t^{q})]^{−1}, and
χH_{∗}(F (R^{q+1}, n); ℤ) = (1 + t^{q})(1 + 2t^{q})....(1 + (n − 1)t^{q}).

Remark: A straightforward geometric interpretation of Proposition 9.1 is given by considering the ordered pairs of distinct pure braids, and using the classical theory of Samelson products in H_{∗}(ΩF (R^{3}, n); ℤ) to measure these differences.

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Department of Mathematics University of Rochester

Rochester, NY 14627

U.S.A

Department of Mathematics, University of Rochester, Rochester, NY 14627

E-mail address: cohf@math.rochester.edu and gitler@math.rochester.edu