On braid groups and homotopy groups
This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2–sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants of pure braids. Natural related questions are posed at the end of this article.
Groups, homotopy and configuration spaces (Tokyo 2005) \conferencestart5 July 2005 \conferenceend11 July 2005 \conferencenameGroups, homotopy and configuration spaces, in honour of Fred Cohen’s 60th birthday \conferencelocationUniversity of Tokyo, Japan \editorNorio Iwase \givennameNorio \surnameIwase \editorToshitake Kohno \givennameToshitake \surnameKohno \editorRan Levi \givennameRan \surnameLevi \editorDai Tamaki \givennameDai \surnameTamaki \editorJie Wu \givennameJie \surnameWu \givennameF R \surnameCohen \urladdrhttp://www.math.rochester.edu/people/faculty/cohf/ \givennameJie \surnameWu \urladdrhttp://www.math.nus.edu.sg/ matwujie \subjectprimarymsc200020F36 \subjectprimarymsc200055Q40 \subjectsecondarymsc200020F40 \subjectsecondarymsc200055U10 \subjectsecondarymsc200018G30 \arxivreference \volumenumber13 \issuenumber \publicationyear2008 \papernumber07 \startpage169 \endpage193 \doi \MR \Zbl \published22 February 2008 \publishedonline22 February 2008 \proposed \seconded \corresponding \version \makeautorefnameexmExample
The purpose of this article is to give an exposition of certain connections between the braid groups (Artin , see Birman ) and classical homotopy groups which arises in joint work of Jon Berrick, Yan-Loi Wong and the authors [9, 10, 3, 38]. These connections emerge through several other natural contexts such as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants of the pure braid groups as developed by T Kohno [23, 24].
The main feature of this article is to identify certain “non-standard” free subgroups of the braid groups via Vassiliev invariants of pure braids. A second feature is to indicate natural ways in which these subjects fit together with classical homotopy theory. This article is an attempt to draw together these connections.
Since this paper was submitted, other connections to principal congruence subgroups in natural matrix groups as well as other extensions have developed. The authors have taken the liberty of adding an additional \fullrefsec:Other.connections with some of these new connections.
Natural related questions are posed at the end of this article.
Although not yet useful for direct computations, there is a strong connection between braid groups and homotopy groups. The braids which naturally arise in this setting also give a large class of special knots and links arising from Brunnian braids as described below as well as by Stanford . It is natural to wonder whether and how these fit together.
The authors take this opportunity to thank Toshitake Kohno, Shigeyuki Morita, Dai Tamaki as well as other friends for this very enjoyable opportunity to participate in this conference. The first author is especially grateful for this mathematical opportunity to learn and to work on mathematics with friends. The authors also thank an excellent referee who gave elegant, useful suggestions regarding the exposition and organization of this article.
2 Braid groups, Vassiliev invariants of pure braids and certain free subgroups of braid groups
The section addresses a naive construction with the braid groups arising as a “cabling” construction. This construction is interpreted in later sections in terms of the structure of braid groups, Vassiliev invariants of pure braids as developed by Toshitake Kohno [23, 24], associated Lie algebras and the homotopy groups of the –sphere (Berrick, Wong and the authors [9, 10, 3, 38]).
Let denote Artin’s –stranded braid group while denotes the pure –stranded braid group, the subgroup of which corresponds to the trivial permutation of the endpoints of the strands. The group is the fundamental group of the configuration space of ordered –tuples of distinct points in the plane
for which for any space .
The group is the fundamental group of the orbit space
obtained from the natural, free (left)-action of the symmetric group on letters . The –stranded braid group of an arbitrary connected surface , , is defined to be the fundamental group of the configuration space of unordered –tuples of distinct points in , . The pure braid group is defined to be the fundamental group
The pure braid groups and are intimately related to the loop space of the 2–sphere as elucidated below in the \fullrefsec:Simplicial objects. Similar properties are satisfied for any sphere as described in \fullrefsec:Other.connections.
To start to address this last point, first consider the free group on letters together with elements for in given by the naive “cabling” pictured in \fullrefbraid below. The braid with in \fullrefbraid is Artin’s generator of . The braids for in \fullrefbraid yield homomorphisms from a free group on letters to
defined on generators in by the formula
The maps are the subject of thew authors’ papers [9, 10] where it is shown is faithful for every . Three natural questions arise: (1) Why would one want to know whether is faithful, (2) are there sensible applications and (3) why is faithful? The answers to these three questions provide the main content of this expository article.
This section addresses one reason why the map is faithful [9, 10]. The method of proof is to appeal to the structure of the Lie algebras obtained from the descending central series for both the source and the target of . The structure of these Lie algebras is reviewed below.
Recall the descending central series of a discrete group given by
where is the subgroup of generated by all commutators
for with . The group is a normal subgroup of with the successive sub-quotients
which are abelian groups having additional structure as follows (cf Magnus, Karrass and Solitar ).
Consider the direct sum of all of the denoted
The commutator function
passes to quotients to give a bilinear map
which satisfies both the antisymmetry law and Jacobi identity for a Lie algebra. (One remark about definitions: The abelian group is both a graded abelian group and a Lie algebra, but not a graded Lie algebra as the sign conventions do not work properly in this context. This situation can be remedied by doubling all degrees of elements in .)
The associated graded Lie algebra obtained from the descending central series for the target yields Vassiliev invariants of pure braids by work of Kohno [23, 24]. This Lie algebra has been used by both Kohno and Drinfel’d  in their work on the KZ equations. The Lie algebra obtained from the descending central series of the free group is a free Lie algebra by a classical result due to P Hall ; see also Serre .
The proof described next yields more information than just the fact that is faithful. The method of proof gives a natural connection of Vassiliev invariants of braids to a classical spectral sequence abutting to the homotopy groups of the –sphere. Sections 5, 6 and 7 below provide an elucidation of this interconnection.
A discrete group is said to be residually nilpotent provided
where denotes the -th stage of the descending central series for . Examples of residually nilpotent groups are free groups, and .
Assume that is a residually nilpotent group. Let
be a homomorphism of discrete groups such that the morphism of associated graded Lie algebras
is a monomorphism. Then is a monomorphism.
If is a free group, and is a monomorphism, then is a monomorphism.
Thus one step in the proof of \fullrefthm: values of theta below is to describe the map
on the level of associated graded Lie algebras
The induced morphism of Lie algebras
satisfies the formula
Examples of this theorem are listed next.
If , then
Thus if , and ,
If , then
Thus if , and ,
Thus if , and ,
To determine the map of Lie algebras with a more global view, the structure of the Lie algebra is useful, and is given as follows. Let denote the free Lie algebra over generated by a set . The next theorem was given in work of Kohno [23, 24], and Falk and Randell .
The Lie algebra is the quotient of the free Lie algebra generated by for modulo the infinitesimal braid relations (also called the horizontal 4T relations or Yang–Baxter–Lie relations)
where denotes the –sided (Lie) ideal generated by the infinitesimal braid relations as listed next:
The maps on the level of associated graded Lie algebras
are monomorphisms. Thus the maps are monomorphisms.
Two remarks concerning are given next.
That is a monomorphism identifies as a free subgroup of rank in . However, there are other, natural free groups of rank in . These arise from the fibrations of Fadell and Neuwirth given by projection maps which delete the -th coordinate and have fibre of the homotopy type of an –fold wedge of circles (Fadell and Neuwirth ).
Let denote the map induced by on the level of fundamental groups. The kernel of is a free group of rank .
The image of has a contrasting feature: Any composite of the natural projection maps precomposed with ,
is a surjection.
The combinatorial behavior of the map is intricate even though the definition is elementary as well as natural. For example, various powers of arise in the computation of the map
for . One example is listed next.
where is independent of the other terms with , and . At first glance, these elements may appear to be ”random“. However, this formula represents a systematic behavior which arises naturally from kernels of certain morphisms of Lie algebras.
The crucial feature which makes the computations effective is the “infinitesimal braid relations”. In addition, the behavior of the map is more regular after restricting to certain sub-Lie algebras arising in the third stage of the descending central series [9, 10]. Finally, the maps also induce monomorphisms of restricted Lie algebras on passage to the Lie algebras obtained from the mod- descending central series [9, 10].
4 Simplicial objects, and –objects
One of the great insights in classical homotopy theory, due to Moore and then ‘extended’ by Kan as well as Rourke and Sanderson is that not only are homology groups a combinatorial invariant, but so are homotopy groups. The basic combinatorial framework is that of a simplicial set developed in [31, 11, 5, 27] and a –set developed in  both of which model the combinatorics of a simplicial complex.
A –set is a collection of sets
with functions, face operations,
which satisfy the identities
if . A –object in a small category is a –set with the given by objects in and the maps given by morphisms in . Thus, a –group is a –set for which all are group homomorphisms.
Let denote the –simplex with the inclusion of the -th face. Assume that each set is given the discrete topology unless otherwise stated. The geometric realization of a –set is given by
where is the equivalence relation generated by requiring that if and , then
Examples of –sets are given next.
A choice of –set with exactly one 0–simplex and one 1–simplex has geometric realization given by the circle.
A natural example of a –group arises from the pure braid groups for a path-connected surface . Define
the -st pure braid group for the surface .
There are homomorphisms
for obtained by deleting the -st strand in . The homomorphisms are induced on the level of fundamental groups of configuration spaces by the projection maps
given be deleting the -st coordinate and satisfy the identities required for a –group.
In case , the associated –group gives basic information about the homotopy groups of the 2–sphere . In case is a closed orientated surface, the –group does not admit the structure of a simplicial group as given in the next \fullrefdefn:simplicial.set.
Simplicial sets are defined next.
A simplicial set is
a –set together with
functions, degeneracy operations,
which satisfy the simplicial identities
A simplicial-object in a small category is a simplicial-set with the given by objects in for which both face maps and degeneracies are given by morphisms in .
Thus, a simplicial-group
is a simplicial-set for which all of the are groups with face and degeneracies given by group homomorphisms.
Two examples of simplicial sets are given next.
The simplicial 1–simplex has two 0–simplices and . The –simplices of are sequences for . All of the non-degenerate simplices are , , and .
The simplicial circle is a quotient of the simplicial 1–simplex obtained by identifying and . There are exactly two equivalence classes of non-degenerate simplices given by , and . Furthermore, the simplicial circle is given in degree by
a single point in case , and
points for in case for which and are identified.
In what follows below, it is useful to label these simplices by
for with .
Homotopy groups are defined for any –set, but the definition admits a simple description for simplicial sets which satisfy an additional condition.
A simplicial set is said to satisfy the extension condition if for every set of –simplices , , …, , , …, which satisfy the compatibility condition
there exists an –simplex such that for . A simplicial set which satisfies the extension condition is also called a Kan complex.
For each Kan complex with base-point and , the -th homotopy set is defined for as the equivalences classes of simplices , denoted , such that
for all , and
two such simplices are equivalent provided there exists a simplex such that , , and for all .
If satisfies the extension condition, and , then is a group. In case , is also abelian.
Basic examples of (i) Kan complexes as well as (ii) –groups which do not admit the structure of a Kan complex are given next.
A simplicial group always satisfies the extension condition as shown in .
If is a simplicial group, then the -th homotopy group of is the quotient group
for which and
Additional, related information is stated next for certain –groups which are not necessarily simplicial groups.
An example of a simplicial group obtained naturally from Artin’s pure braid groups is described next.
Consider –groups with as given in \fullrefexm:delta.set for surfaces . Specialize to the surface
In this case, there are also homomorphisms
obtained by “doubling” the -st strand. The homomorphisms are induced on the level of fundamental groups by the maps for configuration spaces
defined by the formula
where for a point in with
Thus the pure groups in case provide an example of a simplicial group denoted
Consider a pointed topological space . The pointed loop space of , , has a natural multiplication coming from ”loop sum“ which is not associative, but homotopy associative. Milnor proved that the loop space of a connected simplicial complex is homotopy equivalent to a topological group . James  proved that the loop space of the suspension of a connected CW–complex is naturally homotopy equivalent to a free monoid as explained by Hatcher [19, page 282]. Milnor realized that the James construction could be translated directly into the the language of simplicial groups as described next .
Let denote a pointed simplicial set (with base-point and ). Define Milnor’s free simplicial group in degree by
Then is a simplicial group with face and degeneracy operations given by the natural multiplicative extension of those for . In addition, the face and degeneracy operations applied to a generator is either another generator or the identity element.
An example of is given by the simplicial circle. Notice that in degree is isomorphic to the free group by \fullrefexm:basic.simplicial.set.
Milnor defined the geometric realization of a simplicial set for which the underlying topology of is discrete . Recall the inclusion of the -th face togther with the projection maps to the -th face [5, 11, 27].
The geometric realization of is
where denotes the equivalence relation generated by requiring
if and , then and
if and , then
If is a reduced simplicial set (that is is equal to a single point ), then the geometric realization is homotopy equivalent to . Thus the homotopy groups of (as given in  and \fullrefexm:extension.condition) are isomorphic to the homotopy groups of the space .
Consider the special case of . Then the geometric realization is homotopy equivalent to , and there are isomorphisms
5 Pure braid groups of surfaces as simplicial groups and –groups
The homomorphism which arises from the cabling operation described in \fullrefbraid satisfies the following properties.
The homomorphisms give a morphism of simplicial groups
for which the homomorphism is the restriction of to .
By \fullrefthm: embeddings of Lie algebras, the homomorphisms are monomorphisms and so the morphism is a monomorphism of simplicial groups.
There is exactly one morphism of simplicial groups with the property that .
Thus, the picture given in \fullrefbraid is a description for generators of in the simplicial group . These features are summarized next.
The homomorphisms (“pictured” in \fullrefbraid) give the unique morphism of simplicial groups
with . The map is an embedding. Hence the -th homotopy group of , isomorphic to , is a natural sub-quotient of . Furthermore, the smallest sub-simplicial group of which contains the element is isomorphic to .
On the other-hand, the homotopy sets for the –group are also giving the homotopy groups of the 2–sphere via a different occurrence of . The homeomorphism of spaces
for and where denotes a set of there distinct points in is basic for the next Theorem .
If and , then there are isomorphisms
6 Brunnian braids, “almost Brunnian” braids, and homotopy groups
The homotopy groups of a simplicial group, or the homotopy sets of a –group admit a combinatorial description as pointed out in \fullrefexm:extension.condition. These homotopy sets are the set of left cosets where is the group of –cycles and is the group of –boundaries for the –group.
Recall \fullrefexm:delta.set in which the –group is specified by the –stranded pure braid group for a connected surface . The main point of this section is that the –cycles are given by the ”Brunnian braids“ while the –boundaries are given by the “almost Brunnian braids”, subgroups considered next which are also important in other applications (Magnum and Stanford ).
Consider the –stranded pure braid group for any (connected) surface , the fundamental group of . The group of Brunnian braids is the subgroup of given by those braids which become trivial after deleting any single strand. That is,
The “almost Brunnian” –stranded braid group is
The subgroup of consists of those braids which are trivial after deleting any one of the strands , but not necessarily the first.
Consider the simplicial group with
for as given in \fullrefexm:AP.
In this case, notice that that the map is a split epimorphism. Thus the homotopy groups of the simplicial group are all trivial.
An inspection of definitions gives that
the group of –cycles of , , is precisely while
the group of –boundaries, , is exactly .
This feature is recorded next as a lemma.
Let denote a connected surface with associated –group (as given in \fullrefexm:delta.set). Then the following hold.
The group of –cycles is .
The group of boundaries is .
There is an isomorphism
Furthermore, is the trivial group.
There is an isomorphism of left cosets which is natural for pointed embeddings of connected surfaces
Properties of the –group for the –sphere is the main subject of  where the next result is proven.
If and , then
is a group which is isomorphic to the classical homotopy group .
Furthermore, there is an exact sequence of groups
If is a surface not homeomorphic to either or , and , then and are free groups. If is any surface, and , then and are free groups.
One classical example of a Brunnian braid group is which is isomorphic to the principle congruence subgroup of level in as given in \fullrefsec:Other.connections below.
One question below in \fullrefsec:Questions is to consider properties of the free groups obtained from the intersections as well as where is the homomorphism in \fullrefsec:Why.is.Theta.faithful.. These groups are precisely the cycles and boundaries for .
If , then as well as are countably infinitely generated free groups.
The standard Hall collection process or natural variations can be used to give inductive recipes rather than closed forms for generators. T Stanford has given a related elegant exposition of the Hall collection process . The analogous process was applied by Cohen and Levi  to give group theoretic models for iterated loop spaces.
The connection of the homotopy groups of as well as the Lie algebra attached to the descending central series of the pure braid groups is discussed next.
The group is a normal subgroup of . There are isomorphisms
The method of proving that the maps are monomorphisms via Lie algebras admits an interpretation in terms of classical homotopy theory. The method is to filter both simplicial groups and via the descending central series, and then to analyze the natural map on the level of associated graded Lie algebras.
On the other-hand, the Lie algebra arising from filtering any simplicial group by its’ descending central series gives the –term of the Bousfield–Kan spectral sequence for the simplicial group in question . Similarly, filtering via the mod- descending central series gives the classical unstable Adams spectral sequence [5, 11, 38].
Thus the method of proof of \fullrefthm: embeddings of Lie algebras is precisely an analysis of the behavior of the natural map on the level of the –term of the Bousfield–Kan spectral sequence. This method exhibits a close connection between Vassiliev invariants of pure braids and these natural spectral sequences. The next result is restatement of \fullrefthm: embeddings of Lie algebras proven in [9, 10].
The maps on the level of associated graded Lie algebras
are monomorphisms. Thus the maps induce embeddings on the level of the –term of the Bousfield–Kan spectral sequences for .
7 Other connections
Several further connections, outlined next, emerged after this article was submitted.
Connection to principal congruence subgroups
One basic construction above is the Brunnian braid groups . Recently, the authors have proven (unpublished) that the Brunnian braid group is isomorphic to the principal congruence subgroup of level in .
This identification may admit an extension by considering the Brunnian braid groups as natural subgroups of mapping class groups for genus surfaces. The subgroups may embed naturally in via classical surface topology using branched covers of the 2–sphere (work in progress).
Connections to other spheres
The work above has been extended to all spheres as well as other connected CW–complexes . One way in which other spheres arise is via the induced embedding of free products of simplicial groups
The geometric realization of is homotopy equivalent to by Milnor’s theorem stated above as 4.1.
Furthermore, is homotopy equivalent to a weak infinite product of spaces for all .
Connection to certain Galois groups
Consider automorphism groups where is or a completion of given by either the pro-finite completion or the pro- completion . Certain Galois groups are identified as natural subgroups of these automorphism groups by Belyĭ , Deligne , Drinfel’d [13, 14], Ihara [20, 21] and Schneps .
One example is Drinfel’d’s Grothendieck–Teichmüller Galois group