J. Group Theory 15 (2012), 57–83 Journal of Group Theory DOI 10.1515/JGT.2011.095 ( de Gruyter 2012
Braids and crossed modules
J. Huebschmann (Communicated by Jim Howie)
Abstract. For n d 4, we describe Artin’s braid group on n strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations.
1 Introduction
The braid group Bn on n strings was introduced by Emil Artin in 1926, cf. [23] and the references there. It has various interpretations, specifically, as the group of geo- metric braids in R3, as the mapping class group of an n-punctured disk, etc. The group B1 is the trivial group and B2 is infinite cyclic. Henceforth we will take n d 3. The group Bn has n 1 generators s1; ...; sn 1 subject to the relations
sisj ¼ sjsi; ji jj d 2; ð1:1Þ
sjsjþ1sj ¼ sjþ1sjsjþ1; 1 c j c n 2: ð1:2Þ
The main result of the present paper, Theorem 5.2 below, says that, as a crossed module over itself, (i) the Artin braid group Bn has a single generator, which can be taken to be any of the Artin generators s1; ...; sn 1, and that, furthermore, (ii) the kernel of the surjection from the free Bn-crossed module Cn in any one of the sj’s onto Bn coincides with the second homology group H2ðBnÞ of Bn, well known to be cyclic of order 2 when n d 4 and trivial for n ¼ 2 and n ¼ 3. Thus, the crossed module property corresponds precisely to the relations (1.2), that is, these relations amount to Pei¤er identities (see below). When Cn is taken to be the free Bn-crossed s3 1 module in s1, for n d 4, the element s1s1 of Cn has order 2 and generates the copy Z s3 1 of H2ðBnÞ G =2inCn; the element s1s1 of Cn being the generator of the kernel of the projection from Cn to Bn, it is then manifest how this central element forces the relations (1.1). The braid groups are special cases of the more general class of Artin groups [5], [14]. However, for a general Artin group presentation, there does not seem to be an obvious way to concoct a crossed module of the same kind as that for the braid group presentation, and the results of the present paper seem to be special for braid groups. 58 J. Huebschmann
That special feature of ordinary braid groups is presumably related with the fact that Whitehead’s argument establishing the freeness of a crossed module arising from attaching 2-cells to a space involves a homotopy defined over an ordinary disk rather than on a disk with handles. More recently augmented racks [15] and quandles have been explored in the literature, and there is an intimate relationship between augmented racks, quandles, crossed modules, braids, tangles, etc. [15]. See also Remark 5.1 below. Is there a single theory, yet to be uncovered, so that augmented racks, quandles, links, braids, tangles, crossed modules, Hecke algebras, etc. are di¤erent incarnations thereof? In Section 6 below we explain how operads are behind our approach and, perhaps, operads can be used to build such a single theory. We hope our paper contributes to the general understanding of the situation.
2 Historical remarks
The defining relations (1.1) and (1.2) for Bn are intimately connected with Hecke algebra relations. V. Jones discovered new invariants of knots [20] by analyzing cer- tain finite-dimensional representations of the Artin braid group into the Temperley- Lieb algebra [35], that is, a von Neumann algebra defined by a kind of Hecke algebra relations. In the late 1970’s, working on J. H. C. Whitehead’s unsettled problem [38] of whether any subcomplex of an aspherical 2-complex is itself aspherical [19], I had noticed that the relations (1.2) strongly resemble the Pei¤er identities, cf. [28], [29], defining a free crossed module, but it seems that this similarity has never been made explicit in the literature nor has it been used to derive structural insight into braid groups or into related groups such as knot or link groups. It is perhaps worthwhile noting here that the paper [28] was actually submitted for publication as early as June 13, 1944. That kind of similarity gets even more striking when one examines Whitehead’s original proof [38] of the following result of his: Adding 2-cells to a pathwise con- nected space K so that the space L results yields the p1ðKÞ-crossed module p2ðL; KÞ that is free on the 2-cells that have been attached to form L. A precise form of this result will be spelled out below as Theorem 3.1. The crucial step of that proof proceeds as follows: Let B2 denote the 2-disk, S 1 its boundary circle, and denote by o the various base points. To prove that the obvious map from the free p1ðKÞ- crossed module on the 2-cells in LnK to p2ðL; KÞ is injective, Whitehead considers a null-homotopy ðB2 I; S 1 I; o IÞ!ðL; K; oÞ; when this null-homotopy is made transverse, the pre-images of suitable small disks in the interiors of the 2-cells in LnK form a link in the solid cylinder B2 I (more precisely: a partial link based at the 2-disk B2 at the bottom of the cylinder) which truly arises from a kind of tangle or braid; Whitehead then argues in terms of the exterior of that link and in particular interprets the Wirtinger relations corresponding to the crossings in terms of the iden- tities defining a free crossed module, that is, in terms of the Pei¤er identities. In the late 1970’s I learnt from R. Pei¤er (private communication) that Reidemeister had also been aware of the appearance of this kind of link resulting from a null homo- Braids and crossed modules 59 topy. A more recent account of Whitehead’s argument can be found in [7]. See also the survey article [6], where the relationship between identities among relations and links is explained in detail, in particular in terms of the notion of picture, a picture being a geometric representation of an element of a crossed module; a geometric object equivalent to a picture had been exploited already in [28] under the name Randwegaggregat. These links very much look like the tangles that have been used to describe the Temperley-Lieb algebra, cf. [21, p. 8] which, in turn, leads, via a trace, to a descrip- tion of the Jones polynomial. The formal relationship between Pei¤er identities and this kind of link was also exploited in [33]. In the early 1980’s, some of my ideas related with identities among relations and crossed modules merged into the papers [10], [12], and [19]; in particular, pictures are crucially used in these papers.
3 Crossed modules We will denote the identity element of a group by e. Let C and G be groups, q : C ! G a homomorphism, and suppose that the group C is endowed with an action of G which we write as
G C ! C; ðx; cÞ 7! xc; x A G; x A C:
These data constitute a crossed module provided
relative to the conjugation action of G on itself q is a homomorphism of G-groups, and
in C the identities
aba 1 ¼ qab ð3:1Þ
are satisfied. We refer to the identities (3.1) or, equivalently,
aba 1ðqabÞ 1 ¼ e; ð3:2Þ as Pei¤er identities. Morphisms of crossed modules are defined in the obvious way. Thus crossed modules form a category. Let q : C ! G be a crossed module. For later reference, we recall the following facts:
The image qðCÞ J G is a normal subgroup; we will write the quotient group as Q.
The kernel kerðqÞ of q is a central subgroup of C and acquires a Q-module structure.
Via the G-action, the abelianized group CAb acquires a Q-module structure. 60 J. Huebschmann
Let j : Y ! C be a map. The crossed module q : C ! G is said to be free on j provided the following universal property is satisfied: Given a crossed module d : D ! H, a homomorphism a : G ! H, and a (set) map b : Y ! D such that a q j ¼ d b, there is a unique group homomorphism b : C ! D with b j ¼ b such that the diagram
q C? ! G? ? ? ? ? by ya ð3:3Þ d D ! H is a morphism of crossed modules from q to d. When q : C ! G is free on j : Y ! C, in view of the universal property of a free crossed module, the abelianized group CAb is then the free Q-module having (a set in bijection with) Y as a Q-basis. Consequently, the map j is then injective and, somewhat loosely, one says that q : C ! G is free on Y. Given the crossed module q : C ! G, we will refer to C as a G-crossed module; when q : C ! G is a free crossed module, we will refer to C as a free G-crossed module. Let L be a space obtained from the pathwise connected space K by the operation of attaching 2-cells. In [38], J. H. C. Whitehead attempted an algebraic description of the second homotopy group p2ðLÞ. In [39], he reformulated his earlier results in terms of a precise algebraic description of the second relative homotopy group p2ðL; KÞ. In [40], he finally codified the result as follows:
Theorem 3.1 (Whitehead). Via the homotopy boundary map from p2ðL; KÞ to p1ðKÞ, the group p2ðL; KÞ is the free crossed p1ðKÞ-module in the 2-cells that have been attached to K to form L.
Recall that the geometric realization of a presentation hX; Ri of a group is the 2-complex KðX; RÞ with a single vertex whose 1- and 2-cells are in bijection with X and R, respectively, in such a way that, for r A R, the attaching map of the associated 2-cell corresponds to the word xr in the generators X when the relator r is written out as a word in the generators X. Let hX; Ri be a presentation of a group, let F be the free group on X, and let N be the normal closure in F of the relators R.Anidentity among relations [28] (synony- mously: identity sequence) is an m-tuple ðq1; ...; qmÞ (m may vary) ofQ pairs of the j A A G 1 kind qj ¼ðxj; rj Þ where xj F, rj R, j ¼ 1, such that evaluation of xjrj xj in F yields the identity element e of F. Among the observations made in [28], we quote only the following ones: (i) Given two relators r and s, the quadruple
1 1 ððe; rÞ; ðe; sÞ; ðe; r Þ; ðxr; s ÞÞ; ð3:4Þ Braids and crossed modules 61
where xr A F refers to the evaluation of r in F, is always an identity among rela- tions, independently of a particular presentation. In the context of group presen- tations, these identities are referred to in the literature as Pei¤er identities. (ii) The group of identities among relations modulo that of identities of the kind (3.4) is isomorphic to the second homotopy group of the geometric realization of the presentation hX; Ri. Here the term ‘‘second homotopy group’’ is to be interpreted in the language of homotopy chains, which here captures the second homology group of the universal covering space of the geometric realization of the presentation under discussion. (iii) The combinatorial structure of a 3-manifold can be characterized in terms of identities among relations; in particular, the operation of attaching a 3-cell ad- mits an interpretation in terms of killing a certain identity among relations. In [29], some of these results are rediscussed and the idea of a free crossed module is introduced without name. A modern account of identities among relations and crossed modules can be found in [6].
Example 3.2. Given the group G, with respect to the adjoint action, the identity homomorphism G ! G is a crossed module in an obvious manner. In the present paper, we will explore this crossed module for the special case where G is the braid group Bn on n strings.
Example 3.3. The injection of a normal subgroup into a group is a crossed module in an obvious way.
Example 3.4. Given a group G, the obvious map G ! AutðGÞ which sends a member of G to the inner automorphism determined by it is a crossed module. This observa- tion has been explored in [18].
Example 3.5. Given the central extension 1 ! A ! E ! G ! 1 of (discrete) groups, where A is an abelian group, conjugation in E induces a crossed module structure on E ! G.
Example 3.6. Let hX; Ri be the Wirtinger presentation of a knot in R3 with one of the relators discarded, let K be the geometric realization of hX; Ri, pick a generator x A X, and let r be the relator r ¼ x. The normal closure of r is well known to be the entire knot group. Let L ¼ K U e2 be the geometric realization of the resulting pre- sentation hX; R U frgi of the trivial group. By construction, p1ðKÞ is the group of the knot and, in view of Whitehead’s theorem (Theorem 3.1 above), the homotopy boundary map q : p2ðL; KÞ!p1ðKÞ is a free crossed module, the p1ðKÞ-crossed module p2ðL; KÞ being free on the single generator ½r . Since L is actually con- tractible, q : p2ðL; KÞ!p1ðKÞ is an isomorphism. Thus the group of the knot is a free crossed module over itself in a single generator. Suitably interpreted, this kind of observation is also valid for a geometrically indecomposable link. 62 J. Huebschmann
Example 3.7. Let G be a group. A G-rack (or augmented rack [15], or G-automorphic set [4]) is a G-set Y together with a map q : Y ! G such that
qðxyÞ¼xqðyÞx 1; x A G; y A Y: ð3:5Þ
The universal object associated with the G-rack Y is a crossed module q : CY ! G [15] where the notation q is abused. In particular, when Y is the free G-set G S for a set S, the crossed module q : CY ! G is the free crossed module on S. Notice that the identities (3.5) yield again a version of the Pei¤er identities.
4 The main technical result
Theorem 4.1. Let G be a group, let S J G be a subset of G whose normal closure is all of G, let q : C ! G be the free crossed module on S, and suppose that the induced map CAb ! GAb is an isomorphism of abelian groups. Then q fits into a central extension of the kind q 0 ! H2ðGÞ!C ! G ! 1: ð4:1Þ
Proof. Let hX; Ri be a presentation of G, let F be the free group on X, and let N be the normal closure of R in F. Abusing notation, let S J F be a subset which under the projection from F to G is mapped bijectively onto S J G. Then hX; R U Si is a presentation of the trivial group. Let K be the geometric realization of the presentation hX; Ri and L that of the presentation hX; R U Si of the trivial group. By construction, the fundamental group p1ðKÞ of K is G. In view of Whitehead’s theorem (Theorem 3.1 above), the homo- topy boundary map q : p2ðL; KÞ!G is a free crossed module, the G-crossed module C ¼ p2ðL; KÞ being free on the set S. The essential part of the argument comes down to the following:
Proposition 4.2. Under these circumstances, the induced map H2ðKÞ!H2ðLÞ is an isomorphism.
Proof. The group H2ðL; KÞ is free abelian on a basis in bijection with the set S, and inspection of the exact homology sequence
Hq 0 ! H2ðKÞ !H2ðLÞ !H2ðL; KÞ ! H1ðKÞ !0 ð4:2Þ of the pair ðL; KÞ shows that the canonical map H2ðKÞ!H2ðLÞ is an isomorphism if and only if the homology boundary H2ðL; KÞ!H1ðKÞ G H1ðGÞ is an isomor- phism. Recall that, given a pair ðA; BÞ of connected spaces, the relative Hurewicz map induces an isomorphism
~ p2ðB; AÞAb ! H2ðA; BÞ Braids and crossed modules 63
of p1ðBÞ-modules from the abelianization p2ðB; AÞAb of p2ðB; AÞ onto the second relative homology group H2ðA; B~Þ of the pre-image A of A in the universal cover B~ of B modulo that universal cover. Thus the requirement that the induced map CAb ! GAb be an isomorphism of abelian groups says precisely that the homology boundary
H2ðL; KÞ!H1ðKÞ G H1ðGÞ is an isomorphism whence the assertion. r
We can now complete the proof of the theorem. Since L is simply connected, the Hurewicz map p2ðLÞ!H2ðLÞ is an isomorphism and the exact homotopy sequence of the pair ðL; KÞ takes the form
q p2ðKÞ!H2ðLÞ!p2ðL; KÞ!G ! 1: ð4:3Þ
It is a classical fact that the cokernel of the Hurewicz map p2ðKÞ!H2ðKÞ is the second homology group H2ðGÞ of G [17]. In view of Proposition 4.2, the exact homo- topy sequence (4.3) induces the asserted central extension (4.1). r
Example 4.3. Let A be the free abelian group on a set S. Then H2ðAÞ is the exterior square A5A and the free crossed A-module C on S fits into a central extension
q 0 ! A5A ! C ! A ! 1: ð4:4Þ
This extension is the universal central extension of A that corresponds to the universal A5A-valued skew-symmetric 2-cocycle on A; the skew-symmetry determines this 2-cocycle uniquely. In Section 7 we will characterize this kind of central extension by a universal property.
5 Artin’s braid group
The braid group Bn on n strings has n 1 generators s1; ...; sn 1 subject to the relations
sisj ¼ sjsi; ji jj d 2; ð5:1Þ and
sjsjþ1sj ¼ sjþ1sjsjþ1; 1 c j c n 2: ð5:2Þ
Instead of (5.2), we take the relations
1 1 1 sjsjþ1sjsjþ1sj sjþ1 ¼ e: ð5:3Þ 64 J. Huebschmann
Adding the single relation s1 ¼ e plainly kills the entire group, that is, the normal closure of s1 (or any other sj)inBn is all of Bn. For 1 c j c n 2, let 1 1 1 rj; jþ1 ¼ xjxjþ1xjxjþ1xj xjþ1 ^ and let R ¼fr1; 2; r2; 3; ...; rn 2; n 1g. For 1 c j; k c n 1with j < k 1, let
1 1 rj; k ¼½xj; xk ¼xjxkxj xk ; let R~ denote the finite set of these rj; k, and let R ¼ R^ U R~. Then
hX; Ri ¼ hx1; x2; ...; xn 1; r1; 2; r2; 3; ...; rn 2; n 1; r1; 3; ...; rn 3; n 1i ð5:4Þ is a presentation of Bn. We suppose the notation being adjusted in such a way that, for 1 c j c n 1, the generator sj of Bn is represented by xj. Adding the relator r ¼ x1 or, more generally, a relator of the kind r ¼ xj,1c j c n 1, we obtain the presentation hX; R U frgi of the trivial group. Let K ¼ KðX; RÞ be the geometric realization of the presentation hX; Ri and let L ¼ K U e2 be the geometric realization of the presentation hX; R U frgi of the trivial group. By construction, p1ðKÞ is the group Bn. In view of Whitehead’s theorem (Theorem 3.1 above), the homotopy boundary map q : p2ðL; KÞ!Bn is a free crossed module, the Bn-crossed module p2ðL; KÞ being free on the single genera- tor ½r . We will write p2ðL; KÞ as Cn.
Remark 5.1. In the language of Bn-racks, cf. Example 3.7 above, the universal object associated with the Bn-rack Bn fs1g (with the obvious structure) is the free Bn-crossed module Cn on the single generator s1. Thus the universal object associated with the Bn-rack Bn fs1g recovers Bn itself up to the central extension (4.1) by H2ðBnÞ. We shall show in Section 7 below that this central extension is actually univer- sal in a suitable sense.
Theorem 5.2. The kernel of q : Cn ! Bn is a central subgroup and is canonically iso- morphic to H2ðBnÞ. Furthermore, for n ¼ 3, the group H2ðBnÞ is zero and for n d 4, the group H2ðBnÞ is cyclic of order 2, generated as a subgroup of Cn by
s3 1 ½r ½r A Cn: ð5:5Þ
Consequently, as a crossed module over itself, Bn is generated by s1, subject to the relation
s3 s1 ¼ s1: ð5:6Þ
Thus the crossed module property corresponds precisely to the relations (5.2), that is, these relations amount to Pei¤er identities, and the relation (5.6) correspond exactly to the relations (5.1). Braids and crossed modules 65
N.B. As a crossed module over itself, Bn has a single generator, which can be taken to be any of the generators s1; ...; sn 1. In the statement of the theorem, we have written out the relation (5.6) with respect to the generator s1. But we could have written out an equivalent relation in terms of any of the generators s1; ...; sn 1.
Proof. The first statement is a special case of Theorem 4.1. Indeed, Bn is the normal closure of any of the generators sj and, cf. the proof of Proposition 4.2 above,
Z ðCnÞAb G p2ðL; KÞAb G H2ðL; KÞ!H1ðKÞ G H1ðBnÞ G is plainly an isomorphism. The group B2 is infinite cyclic and hence has zero second homology group. The group B3 is well known to be isomorphic to the trefoil knot group and hence has zero second homology group as well; indeed, any knot complement is a homology circle. Henceforth we suppose that n d 4. We will now derive the relation (5.6). As usual we denote the 1-skeleton of the cell complex K by K 1. The exact homotopy sequence of the pair ðL; K 1Þ takes the form
1 qL 1 0 ! p2ðLÞ!p2ðL; K Þ!p1ðK Þ!1: ð5:7Þ
1 Since the p1ðK Þ-action on p2ðLÞ factors through p1ðLÞ which is the trivial group, the 1 p1ðK Þ-action on p2ðLÞ is trivial, and the group extension (5.7) is necessarily central. 1 The second homotopy group p2ðLÞ lies in p2ðL; K Þ as the group of (classes of ) iden- tities among relations. 1 1 Let F G p1ðK Þ denote the free group on X. Keeping in mind that p2ðL; K Þ is the free F-crossed module on R U frg, let
1 I1; 3 ¼ððx3; rÞ; ðe; r Þ; ðe; r1; 3ÞÞ: ð5:8Þ
Inspection shows that
1 1 qLð½I1; 3 Þ ¼ e A F G p1ðK Þ whence ½I1; 3 A p2ðLÞ J p2ðL; K Þ:
Indeed, I1; 3 is plainly an identity among relations. 1 Consider the canonical projection from p2ðL; K Þ to Cn G p2ðL; KÞ. Since under this projection each generator in R goes to e A p2ðL; KÞ, the image ½I1; 3 A p2ðL; KÞ 1 of the member ½I1; 3 of p2ðLÞ J p2ðL; K Þ can plainly be written as
s3 1 ½I1; 3 ¼ ½r ½r A p2ðL; KÞ G Cn: ð5:9Þ
Here the bracket notation ½ is slightly abused. Since the 2-complex L is simply connected, the Hurewicz map p2ðLÞ!H2ðLÞ is an isomorphism. Furthermore the reasoning in the proof of Theorem 4.1 shows that the canonical map H2ðKÞ!H2ðLÞ 66 J. Huebschmann is an isomorphism. Hence, in the exact homotopy sequence (5.7), we can replace p2ðLÞ with H2ðKÞ. Inspection of the commutative diagram
1 qL 0 ! H2?ðKÞ ! p2ðL?; K Þ ! F? ! 1 ? ? ? ? ? ? y y y ð5:10Þ q 0 ! H2ðBnÞ ! Cn ! Bn ! 1 reveals that ½I1; 3 A H2ðBnÞ. Likewise, for 1 c j; k c n 1 with k j d 2, corresponding to the relator
1 1 rj; k ¼½xj; xk ¼xjxkxj xk ;
1 we can construct an identity among relations Ij; k such that p2ðLÞ J p2ðL; K Þ is the free abelian group in
1 ½I1; 3 ; ...; ½In 3; n 1 A p2ðLÞ J p2ðL; K Þ:
For completeness, we will now make these identities among relations explicit even though we do not strictly need the explicit form of these identities: For 4 c k c n 1, let
1 I1; k ¼ððxk; rÞ; ðe; r Þ; ðe; r1; kÞÞ: ð5:11Þ
Next, for 2 c j c n 1, add the relator rj ¼ xj and, for 2 c j; k c n 1 with k j d 2, let ^ 1 Ij; k ¼ððxk; rjÞ; ðe; rj Þ; ðe; rj; kÞÞ: ð5:12Þ
This is an identity sequence for the enlarged presentation. However. making suitable substitutions, we can transform this identity sequence into one over the original pre- 1 1 1 sentation. For example, for j ¼ 2, making the substitutions r1; 2 x1x2rx2 x1 for the relator r2 in I^2; k, we obtain the identity sequence
1 1 I2; k ¼ððxk; r1; 2Þ; ðxkx1x2; rÞ; ðx1x2; r Þ; ðe; r1; 2Þ; ðe; r2; kÞÞ: ð5:13Þ
In the same way, we can construct identity sequences Ij; k for the remaining cases. However we do not strictly need these identities among relations explicitly: The rank of the free abelian group p2ðLÞ G H2ðLÞ G H2ðKÞ equals the cardinality of the set R~. Indeed, let K^ ¼ KðX; R^Þ be the geometric realization of the presentation hX; R^i. The canonical map H2ðKÞ!H2ðK; K^Þ is easily seen to be an isomorphism of abelian groups, the target group H2ðK; K^Þ being free abelian in the set of 2-cells in KnK^ or, equivalently, in the set R~. The association
Ij; k 7! rj; k ð1 c j; k c n 1; j < k 1Þð5:14Þ Braids and crossed modules 67 induces the composite of the isomorphism p2ðLÞ!H2ðKÞ with the isomorphism from H2ðKÞ to H2ðK; K^Þ. For n d 4, the group H2ðBnÞ is well known to be cyclic of order 2, cf. [11]. We shall reprove this fact in Section 6 below. Furthermore, the left-hand map H2ðKÞ!H2ðBnÞ in the diagram (5.10) is surjective by construction. For n ¼ 4, the group H2ðKÞ G Z is free abelian in the relator r1; 3, and that map takes the form of a surjection
Z ! H2ðB4Þ; r1; 3 7!½I1; 3 : ð5:15Þ
Consequently ½I1; 3 A H2ðB4Þ is non-zero and, the obvious map H2ðB4Þ!H2ðBnÞ being an isomorphism, we conclude that ½I1; 3 A H2ðBnÞ is non-zero for any n d 4. Finally, the explicit description (5.9) of the element ½I1; 3 A Cn justifies at once the relation (5.6) in Theorem 5.2. These observations complete the proof of Theorem 5.2. r
Remark 5.3. The subgroup of Bn (n d 4) generated by x1 and x3 is well known to be free abelian of rank 2; it is convenient to think of this group as the direct product B2 B2, viewed as a subgroup of Bn. The geometric realization Kðx1; x3; r1; 3Þ of the presentation hx1; x3; r1; 3i of B2 B2 is an ordinary torus which is plainly aspherical. Furthermore, hx1; x3; x1; r1; 3i is then a presentation of the free cyclic group C gener- ated by x3; the geometric realization Kðx1; x3; x1; r1; 3Þ thereof is a spine for a solid torus having fundamental group C, the identity among relations I1; 3 is defined over the presentation hx1; x3; x1; r1; 3i of C and, indeed, records the attaching map for a 3-cell to form the solid torus from the spine. Playing the same game as in the pre- vious proof, but with the geometric realizations Kðx1; x3; r1; 3Þ and Kðx1; x3; x1; r1; 3Þ rather than with K and L, respectively, we find that p2ðKðx1; x3; x1; r1; 3ÞÞ is the free C-module freely generated by ½I1; 3 ; the induced map from H2ðKðx1; x3; r1; 3ÞÞ to H2ðKðx1; x3; x1; r1; 3ÞÞ is an isomorphism, both groups are infinite cyclic, the group H2ðKðx1; x3; r1; 3ÞÞ amounts to the second homology group H2ðB2 B2Þ G Z of B2 B2, and under the Hurewicz map from p2ðKðx1; x3; x1; r1; 3ÞÞ to H2ðKðx1; x3; x1; r1; 3ÞÞ the free C-module generator ½I1; 3 goes to a generator of H2ðKðx1; x3; x1; r1; 3ÞÞ. Thus the surjection (5.15) comes down to the map
H2ðB2 B2Þ!H2ðB4Þð5:16Þ induced by the injection of B2 B2 into B4. Hence the map (5.16) yields the gen- erator of H2ðB4Þ and, more generally, of H2ðBnÞ for n d 4. This observation goes back at least to [11] where it is pointed out that this kind of construction, suitably extended, yields the entire integral homology of Bn. The universal covering space of Kðx1; x3; x1; r1; 3Þ is an infinite cylinder together with infinitely many 2-disks attached to the cylinder, more precisely, when the cyl- inder is realized in ordinary 3-space, the disks are attached to the interior of the cylinder. Thus the identity among relations I1; 3 over the presentation hx1; x3; x1; r1; 3i 68 J. Huebschmann of the free cyclic group C is realized geometrically as a tesselated sphere arising from the boundary surface of a solid cylinder having top and bottom a disk labelled r ¼ x1 and wall (lateral surface) labelled r1; 3; the cyclinder wall arises from glueing a rect- angle labelled r1; 3 along the edge labelled x3, and the sphere arises from glueing the top and bottom disks along the circles labelled x1. Indeed, the universal covering space of the spine Kðx1; x3; x1; r1; 3Þ arises from glueing infinitely many copies of that tesselated sphere along the top and bottom disks. That tesselated sphere can be seen as a cycle representing the generator of H2ðBnÞ.
6 Parallelohedral geometry and identities among relations Using the language of identities among relations, we will now construct, under the circumstances of Theorem 5.2, a complete system of p1ðKÞ-module generators of p2ðKÞ and thereafter describe the induced map p2ðKÞ!p2ðLÞ G H2ðKÞ so that the cokernel H2ðBnÞ G Z=2 thereof becomes explicitly visible. We will in particular exploit the geometry that underlies the hyperplane arrangements associated with the root systems Aj ( j d 1). This will provide additional geometric insight into the braid group Bn and will in particular explain why the group H2ðBnÞ is cyclic of order 2 and how the relation (5.6) arises. It will also make our reasoning self-contained. At the referee’s request, we will in particular explain how a classifying space for Bn arises from the permutahedron written below as Pn.
6.1 The configuration space of n particles in the real plane. At the turn of the 19-th century, Hurwitz, Fricke and Klein constructed the mapping class group of the eucli- dean plane with n-punctures and thereby developed a description of the braid group as a fundamental group as follows, cf. [23]: Identify the n-th symmetric power Sn½C of the complex line C with the space of normalized complex degree n polynomials n n 1 z þ a1z þ þan; this space is stratified according to the multiplicities of the roots of the polynomials, the top stratum being the space of polynomials having only single roots, and the braid group Bn is the fundamental group of the top stra- 2 tum. That top stratum coincides with the orbit space FnðE Þ=Sn of the configuration 2 2 space FnðE Þ of n particles moving in the plane E and the symmetric group Sn acts 2 freely on FnðE Þ. In [16], Fox and Neuwirth rediscovered and generalized the descrip- 2 tion of the braid group Bn as the fundamental group of the orbit space FnðE Þ=Sn.In particular, Fox and Neuwirth constructed an Sn-equivariant decomposition of the 2 configuration space FnðE Þ into convex cells and showed that the resulting presenta- tion of the fundamental group of the orbit space amounts to the Artin presentation of 2 the braid group Bn. They also showed that the orbit space FnðE Þ=Sn is aspherical, i.e., a classifying space for Bn. Their decomposition into convex cells actually yields a poset whose geometric realization is Sn-equivariantly homotopy equivalent to the 2 configuration space FnðE Þ, see Subsection 6.4 below. 2 With modern technology, the asphericity of the configuration space FnðE Þ is 2 2 straightforward: The space F1ðE Þ is the plane E itself and, for n d 2, forgetting 2 2 one of the particles, we obtain a fibration FnðE Þ!Fn 1ðE Þ with fiber the plane 2 with n 1 punctures. A straightforward induction reveals that FnðE Þ is aspherical Braids and crossed modules 69
2 whence so is the orbit space FnðE Þ=Sn. This is the argument used in [16]; the result established in that reference is actually more general.
6.2 Permutahedra. A parallelohedron is a convex polyhedron whose translates by a lattice have disjoint interior and cover n-dimensional real a‰ne space. This notion is implicit in [27] and was introduced in [36] and [37]. See [41, p. 17 ¤.] for detailed information. In dimension 2, the only parallelohedra are the parallelogram and the hexagon. In dimension 3, there are exactly five parallelohedra: the cube, hexagonal prism, truncated octahedron, rhombic dodecahedron, and the elongated rhombic dodecahedron. n Let n d 2. Let the symmetric group Sn on n letters act on R via permutation of coordinates. The permutahedron Pn of order n is the convex hull of the Sn-orbit of a generic point, e.g. of the point e ¼ð1; 2; ...; nÞ of Rn; in that case, relative to n the standard coordinates x1; ...; xn on R , the permutahedron Pn lies in the a‰ne ðn 1Þ-dimensional plane in Rn with equation X nðn þ 1Þ x ¼ : i 2
The 1-skeleton of the permutahedron Pn is the Cayley complex of Sn relative to the standard generators; here the convention is that any generator that is an involution is represented by a single unoriented edge rather than by a pair of oppositely oriented edges. Permutahedra were explored already in [32] (not under this name which seems to have been introduced much later). A permutahedron is a special kind of parallelohe- dron. Permutahedra are explored e.g. in [13, p. 65/66]. We take P1 to be a point; the permutahedron P2 is a closed interval, the permutahedron P3 is a hexagon, and the permutahedron P4 is the truncated octahedron, that is, the Archimedean solid having as faces eight hexagons and six squares, the corresponding tesselation of real a‰ne 3-space being the bitruncated cubic honeycomb. In general, Pn is an omnitrun- cated ðn 1Þ-cell. An ordered partition ðQ1; Q2; ...; QrÞ of f1; 2; ...; ng (1 c r c n) is a decomposi- tion into disjoint subsets Qk of f1; 2; ...; ng where the order ðQ1; Q2; ...; QrÞ is part of the structure. The set of ordered partitions of n is partially ordered by refinement: The partial order < is generated by
ðQ1; Q2; ...; QrÞ < ðQ1; ...; Qk U Qkþ1; ...; QrÞ:
As a cellular complex, the permutahedron Pn of order n is the geometric realiza- tion of the poset of ordered partitions of f1; 2; ...; ng, cf. e.g. [24, p. 100]. In terms of the above realization of the permutahedron Pn of order n as the convex hull of the Sn-orbit of a generic point, the cell corresponding to ðQ1; Q2; ...; QrÞ and denoted again ðQ1; Q2; ...; QrÞ is the convex hull of the orbit of the generic point relative to the subgroup of Sn leaving the decomposition fixed. The face 70 J. Huebschmann
ðQ1; Q2; ...; QrÞ has dimension n r. Thus, the permutahedron Pn has a unique maximal element ðf1; 2; ...; ngÞ (of dimension n 1) and n! minimal elements (vertices) ðfsð1Þg; ...; fsðnÞgÞ as s ranges over Sn. For example, the hexagon P3 has the six vertices ðf jg; fkg; flgÞ as ð j; k; lÞ ranges over the permutations of ð1; 2; 3Þ, the six edges
ðf12g; f3gÞ; ðf13g; f2gÞ; ðf23g; f1gÞ; ðf1g; f23gÞ; ðf2g; f13gÞ; ðf3g; f12gÞ; and the single face ðf123gÞ, see [24, Figure 2 on p. 98] for the poset structure of P3. The geometric realization of the truncated octahedron P4 is portrayed in [24, p. 98, Figure 3]. The notation in that reference is ð j1; j2; ...; jrÞ rather than our ðQ1; Q2; ...; QrÞ. For a finite set Q we will write its number of elements as jQj. Given the ordered partition ðQ1; Q2; ...; QrÞ of f1; 2; ...; ng (1 c r c n), we will use the notation
ð j1; j2; ...; jrÞ¼ðjQ1j; jQ2j; ...; jQrjÞ for the associated ordered decomposition
n ¼ j1 þ j2 þ þ jr; of the integer n into a sum of integers jk d 1(1c k c r); we will refer to ð j1; j2; ...; jrÞ as the type of ðQ1; Q2; ...; QrÞ as well as of the corresponding face of Pn, and we will refer to r as the length of the ordered decomposition ð j1; j2; ...; jrÞ of n; similarly as before, the order ð j1; j2; ...; jrÞ is part of the structure of ordered decomposition of n. In [2, Definition 2.5], the terminology ‘‘ordered partition’’ rather than ordered decomposition is used but that usage of the term ordered partition is not consistent with our usage thereof. Given the ordered decomposition ð j1; ...; jrÞ J of n, let Sð j1;...; jrÞ ¼ Sj1 Sjr Sn. The system of ordered decompositions of n is likewise partially ordered by refine- ment; thus, when we denote the order relation by c:
ði1; ...; irÞ c ð j1; ...; jsÞ provided that there is an ordered decomposition s ¼ k1 þ þkr of s such that
i1 ¼ j1 þ þ jk1
i2 ¼ jk1þ1 þ þ jk1þk2 ...
ir ¼ jk1þ þkr 1 þ þ jk1þ þkr :
K Thus ði1; ...; irÞ c ð j1; ...; jsÞ if and only if Sði1;...; irÞ Sð j1;...; jsÞ. We denote the poset of ordered decompositions of n by On. Braids and crossed modules 71
Let } be a general poset. Given p A }, define the sub-poset }dp ¼fx A }; x d pg; op the sub-posets }>p, }cp, }
Sð j1;...; jsÞ and any such subgroup has a unique member T of Sn as its generating set, cf. [3, Chapter 9], where this is established for general Coxeter systems; moreover, the resulting bijection Sn ! On reverses the partial order, that is, induces an isomor- phism
op Sn ! On of posets. We now list a few properties of the permutahedron. These are now more or less obvious, cf. [2, 2.11] or [8, Subsection 6.2]; in [8, p. 548], the permutahedron Pn is referred to as the Zilchgon and written as Cðn 1Þ:
Let ð j1; j2; ...; jrÞ be an ordered decomposition of n. Define shuffð j1; j2; ...; jrÞ as the set that consists of those s A Sn so that sð jÞ < sðkÞ whenever j < k belong to the same block in the decomposition. The complete set of faces of Pn is indexed by ordered pairs consisting of an ordered decomposition ð j1; j2; ...; jwÞ of n and a
ð j1; j2; ...; jwÞ-shu¿e. Such a face is isomorphic to a product Pj1 Pjw of lower dimensional permutahedra and has dimension n w.
The poset of ordered partitions of f1; ...; ng decomposes into types and the induced Sn-action on the poset of ordered partitions is transitive on each type. The faces of Pn or, equivalently, ordered partitions of f1; ...; ng of type ð j1; ...; jrÞ are in
bijection with the system Sn=Sð j1;...; jrÞ of cosets sSð j1;...; jrÞ as s ranges over Sn via
the assignment to sSð j1;...; jrÞ of the convex hull of the points tðeÞ as t ranges over
sSð j1;...; jrÞ. The Sn-action on the faces of Pn of type ð j1; ...; jrÞ is given by the
induced left Sn-action on Sn=Sð j1;...; jrÞ.
In terms of the identification of the complete set of faces of Pn with the disjoint union
6 Sn=Sð j1;...; jrÞ; n¼ j1þ j2þ þ jr
the partial order of ordered partitions of n corresponds to the partial order < 0 defined by ‘s Sði1;...; irÞ < sSð j1;...; jsÞ’ whenever I (i) ði1; ...; irÞ < ð j1; ...; jsÞ (or, equivalently, Sði1;...; irÞ Sð j1;...; jsÞ), and 1 0 A (ii) s s Sði1;...; irÞ. We will now briefly explain, in the case under discussion, that is, for the Coxeter system associated with the Artin braid group on n strands, what is referred to in the literature as the Salvetti complex [30], [31]. We follow the description in [9]. 72 J. Huebschmann
The obvious map
6 Sn On ! Sn=Sð j1;...; jrÞ; ðs; ð j1; ...; jrÞÞ 7! sSð j1;...; jrÞ n¼ j1þ j2þ þ jr is plainly surjective. To understand this map, pick a type ð j1; ...; jrÞ, i.e. ordered decomposition j1 þ þ jr ¼ n of n; then that surjection restricts to the canonical projection Sn ! Sn=Sð j1;...; jrÞ, and the Sn-action permutes the faces of the permutahe- dron having type ð j1; ...; jrÞ. 0 Define a partial order < on Sn On by ‘ðs ; Sði1;...; irÞÞ < ðs; Sð j1;...; jsÞÞ’ whenever 0 1 0 1 0 A s Sði1;...; irÞ < sSð j1;...; jsÞÞ and, furthermore, lðs s Þ < lðts s Þ for every t Sði1;...; irÞ; here l refers to the word length relative to the Artin generators. This yields a partial order on Sn On, and we denote the geometric realization of the resulting poset f by Pn. More precisely, consider the derived complex (or order complex) associated with the poset Sn On. We recall that, for any poset, the derived complex (the chains of members of the poset) is an abstract simplicial complex, and its geometric realiza- tion is a simplicial complex, a polyhedron when the set underlying the poset has finitely many elements. In the case at hand, the resulting polyhedron is cellulated by Coxeter cells in such a way that the resulting cell complex has one cell for each element of Sn On, cf. [9, Lemma 2.2.1]. We note, for completeness, that the partial order in [9] is opposite to the order we are using here. The derived complex of the poset of ordered decompositions of f1; ...; ng has likewise a geometric realization cellulated by Coxeter cells; these are precisely the cells of the permutahedron Pn.By construction, the canonical map
6 Sn On ! Sn=Sð j1;...; jrÞ n¼ j1þ j2þ þ jr is order preserving; since the Coxeter cells correspond, that map induces a unique f cellular map from Pn to Pn. By construction, the group Sn acts freely and cellularly f f on Pn. In the literature, the quotient complex Pn=Sn is referred to as the Salvetti f complex. A variant of the Sn-cell complex Pn occurs already in [25], see Subsection 6.4 below f Proposition 6.1. The orbit space Pn=Sn is a classifying space for Bn.
This is a major result, spread over various places in the literature; we label it ‘‘Proposition’’ since it is not among the results of the present paper. In Subsection 6.3 below we will sketch a proof. f f The projection from Pn to Pn=Sn factors as f f Pn ! Pn ! Pn=Sn; thus the operation of gluing or identifying faces of the permutahedron Pn according f to the Sn-action on Pn or, equivalently, for each type ð j1; ...; jrÞ, identifying faces Braids and crossed modules 73
f in the coset Sn=S to a single cell, yields the cell complex Pn=Sn, and this cell ð j1;...; jrÞ f complex is a classifying space for Bn. By construction, the 2-skeleton of Pn=Sn is the geometric realization KðX; RÞ of the Artin presentation (5.4) of Bn. Immediate consequences of these observations are: f Proposition 6.2. Let n d 3. For 1 c r c n, the ðn rÞ-dimensional cells of Pn=Sn are in bijective correspondence with the ordered decompositions of n of length r. In particular: (i) The 1-cells are in bijection with the n 1 ordered decompositions of n of length n 1. These correspond to the Artin generators s1; ...; sn 1 in the Artin presenta- tion (5.4). (ii) The 2-cells are in bijection with the ordered decompositions of n of length n 2. Those ordered decompositions ð j1; ...; jn 2Þ where jk ¼ 3 for some k (1 c k c n 2) arise from a hexagonal face of Pn (i.e. from a copy of P3) and are in bijection with the n 2 relators in the Artin presentation (5.4) that are ^ members of R; those where jk1 ¼ jk2 ¼ 2 for some k1 < k2 (1 c k1 < k2 c n 2) arise from a square face of Pn (i.e. from a copy of P2 P2) and are in bijection with the relators in the Artin presentation (5.4) that are members of R~. (iii) When n d 4, the 3-cells are in bijection with the ordered decompositions of n of length n 3.
(iv) When n d 4, the ordered length n 3 decompositions ð j1; ...; jn 3Þ where jk ¼ 4 for some k (1 c k c n 3) arise from a face of Pn that is a copy of a truncated octahedron (i.e. from a copy of P4).
(v) When n d 5, the ordered length n 3 decompositions ð j1; ...; jn 3Þ where jk1 ¼ 2,
jk2 ¼ 3 for some k1; k2 (1 c k1; k2 c n 3) arise from a prismatic face of Pn that is a copy of P2 P3.
(vi) When n d 6, the ordered length n 3 decompositions ð j1; ...; jn 3Þ where
jk1 ¼ jk2 ¼ jk3 ¼ 2, for some k1 < k2 < k3 (1 c k1 < k2 < k3 c n 3), arise from a cubic face of Pn that is a copy of P2 P2 P2. f Proposition 6.3. The cellular boundary operator of the cellular chain complex of Pn f as well as that of Pn=Sn is given by the operation of suitably further partitioning the products of permutahedra occuring as faces of the permutahedron Pn.
6.3 Hyperplane arrangements. A hyperplane arrangement H in Rn (also referred to as a real hyperplane arrangement) is a finite family of hyperplanes in Rn, cf. e.g. [41, p. 193]; a complex hyperplane arrangement is defined accordingly. A hyperplane arrangement in Rn decomposes the ambient space Rn into a finite number of convex regions, referred to as chambers. The Weyl group is the group generated by the reflec- n tions in a chosen chamber. Given a hyperplane arrangement H in R , let HC denote the corresponding hyperplane arrangement in Cn and let Y ¼ Cnn6 H. The H A HC Weyl group W acts freely on Y; when W is finite, the orbit space Y=W is a classify- ing space for the Artin group (generalized braid group) associated with the hyper- 2 plane arrangement [14]. The asphericity of the orbit space FnðE Þ=Sn reproduced 74 J. Huebschmann above is a special case thereof. Given a root system—for our purposes knowledge of the Lie algebraic root systems Aj ( j d 1) will su‰ce—, the hyperplanes orthogonal to the roots define a hyperplane arrangement. Given a real hyperplane arrangement we will refer to the convex hull A of a Weyl group orbit of a point in the interior of a chamber as a Coxeter polytope. In the literature, the dual complex thereof is referred to as the Coxeter complex (associated with the corresponding Coxeter system). Let n d 2. The braid group Bn is the Artin group associated with the root system (equivalently: Coxeter graph) An 1. In this case, the Coxeter polytope An 1 is exactly the permutahedron Pn and, as pointed out above, the operation of identifying faces of the Coxeter polytope via the f Sn-action on Pn explained in Subsection 6.2 above yields a cell complex whose 2-skeleton is the geometric realization KðX; RÞ of the Artin presentation (5.4) of Bn. Let H be the hyperplane arrangement in Rn associated with the root system An 1 or, equivalently, with the order n braid arrangement. As before, let Y ¼ Cnn 6 H, and suppose that the arrangement is associated to some Cox- H A HC f eter group W. According to a result in [30], the complex Pn embeds Sn-equivariantly into Y as a deformation retract and hence has the same equivariant homotopy type as Y. In view of the special case of Deligne’s result [14] established in [16], the cell f f complex Pn is aspherical; hence the orbit space Pn=Sn is aspherical as well and thence a classifying space for the associated Artin group Bn. These observations establish Proposition 6.1 above. For a general Artin group associated to a finite Coxeter group W, the construction in [31] yields a classifying space arising from a Coxeter polytope in much the same way; this classifying space construction is also lurking behind the approach in [34].
6.4 Operadic description. As pointed out in [1], [2], the above constructions unify in the language of operads. Let L be the category of (non-empty) finite sets with injective maps as morphisms. Given a category C,apreoperad with values in C is a functor P : Lop ! C [2, Definition 1.1], [24, Definition 2.6 p. 100]. An operad is a preoperad with additional structure (not spelled out here). Rk For each k d 1, the collection of configuration spaces fFnð Þgn constitutes a topological preoperad F ðkÞ. Likewise the construction J ðkÞ (k d 1) [25] yields approx- imations to the functors WkS k and, as noted in [1], [2], each J ðkÞ is a preoperad; the preoperad J ðkÞ is defined to be the collection
ðkÞ k 1 Jn ¼ððPnÞ SnÞ=@ ðn d 0Þ where @ denotes a certain equivalence relation not reproduced here for general k, cf. e.g. [2, Definition 2.12]. Associated with such a preoperad O is a certain partially ordered preoperad KðOÞ where the notation K refers to the complete graph operad [2, Section 1]. We will now recall the equivalence relation for k ¼ 1; this is the case of interest for us and, to reconcile the construction with our description of the various posets given earlier, we interchange the order of Pn and Sn: The faces of the permutahedron Pn of order n are indexed by ordered partitions ðQ1; Q2; ...; QrÞ of f1; 2; ...; ng. Now, Braids and crossed modules 75 given the cell indexed by ðQ1; Q2; ...; QrÞ, let t A Sn be the ðjQ1j; jQ2j; ...; jQrjÞ- shu¿e that corresponds to ðQ1; Q2; ...; QrÞ; for any point p of Pn that belongs to the cell indexed by ðQ1; Q2; ...; QrÞ, let
1 1 ðs; t pÞ @ ðst ; pÞ; s A Sn:
This yields an equivalence relation on Sn Pn, and the quotient modulo the equi- f valence relation @ entails another description of the complex Pn. For example, for the unique maximal cell ðQÞ¼ðf1; 2; ...; ngÞ of dimension n 1, the associated ðjQjÞ-shu¿e is the trivial permutation and the equivalence relation has no e¤ect, that is, the construction produces n! distinct cells of dimension n 1. Likewise, for s A Sn, given the vertex p ¼ðfsð1Þg; ...; fsðnÞgÞ of Pn,
0 1 0 1 0 ðs ; s pÞ @ ðs s ; pÞ; s A Sn; that is, there result n! vertices, i.e. the construction does not produce any new vertices. f The present description also shows that Pn embeds into Pn as a cellular subcomplex. This makes the operation of gluing faces of Pn to arrive at the classifying complex f Pn=Sn even more explicit as the composite f f Pn ! Pn ! Pn=Sn:
ð2Þ Let n d 2. The poset KðFÞn arises from the Fox-Neuwirth decomposition 2 of FnðE Þ into convex cells mentioned in Subsection 6.1 above whereas the poset ð2Þ f KðJÞ yields the cell complex written above as Pn and hence leads to the Salvetti n f complex for the n-strand braid arrangement (written above as Pn=Sn). By a result in [1], the configuration preoperad and the Milgram preoperad are homotopy equi- ð2Þ valent. Hence, in view of [2, Theorem 2.14], the realization of the poset KðJÞn has 2 the Sn-equivariant homotopy type of the configuration space FnðE Þ; we note, as a side remark, that the actual statements in [1] and [2] are more general. The upshot of this observation is that the Fox-Neuwirth decomposition and the Salvetti complex are intimately related in a very precise way which, with a grain of salt, amounts to the operation of passing from a cell decomposition to the dual decomposition. In partic- ular, as pointed out in [1, Remarque 3.12], the Fox-Neuwirth cells are in bijective correspondence with the Salvetti cells.
6.5 Identities among relations. For ease of the reader, we note that the method of pictures developed in [6], [12] and [19] yields a straightforward procedure for reading o¤ the identity sequences (6.2), (6.5), (6.8) and (6.10) below. We have already noted that the association
Ij; k 7! rj; k ð1 c j; k c n 1; j < k 1Þ induces the composite
p2ðLÞ!H2ðK; K^Þð6:1Þ 76 J. Huebschmann of the isomorphism p2ðLÞ!H2ðKÞ with the isomorphism from H2ðKÞ to H2ðK; K^Þ, cf. (5.14) above. For n ¼ 2 and n ¼ 3, the geometric realization K ¼ KðX; RÞ of the Artin presenta- tion (5.4) of Bn is aspherical.
Proposition 6.4. Let n d 4.AsaBn-module, p2ðKÞ has a complete system of generators
A 1 ½Jj; k; l p2ðKÞ J p2ðK; K Þ; 1 c j < k < l c n 1; n 1 given by 3 suitable identities Jj; k; l among the relations of the Artin presentation (5.4) of Bn. Furthermore, the composite of the induced map from p2ðKÞ to p2ðLÞ with (6.1) is given by the associations
½Jj; jþ1; jþ2 7! 2rj; jþ2; ½Jj; jþ1; k 7! rjþ1; k rj; k; ½Jj; k 1; k 7! rj; k rj; k 1; in particular, that map is zero on generators of the kind ½Jj; k; l with k j d 2 and l k d 2. Consequently H2ðBnÞ is cyclic of order 2 and each free generator of H2ðKÞ corresponding to a generator of H2ðK; K^Þ of the kind rj; k for k j d 2 goes to the single non-zero element of H2ðBnÞ. We begin with the preparations for a formal proof of this proposition: Let n d 4 and consider the Artin presentation (5.4) of Bn. The possible 3-dimensional parallelo- hedra that can arise from the full subgraphs of An 1 with three nodes are the trun- cated octahedron or permutahedron P4 (Coxeter graph A3), the hexagonal prism or product P2 P3 of two lower-dimensional permutahedra (Coxeter graph A1 A2), and the cube or product P2 P2 P2 of three lower-dimensional permutahedra (Coxeter graph A1 A1 A1). These correspond to the possible 3-dimensional faces of the permutahedron Pn (n d 4), cf. the discussion in Subsection 6.2 above. We will now realize each of these parallelohedra by an identity among relations. We will proceed for general n d 4: To realize the truncated octahedron we note that, for each 1 c j c n 3, the sub- group of Bn generated by xj, xjþ1, xjþ2 is a copy of B4. Accordingly, consider the identity
Jj; jþ1; jþ2 ¼ðq1; q2; q3; q4; q5; ...; q13; q14Þð6:2Þ among relations arising from reading o¤ an identity sequence from a truncated octa- hedron, each hexagon being labelled with the relator rj; jþ1 or rjþ1; jþ2 and each square being labelled with the relator rj; jþ2:
1 1 1 1 q1 ¼ðxj xjþ1xjþ2xj ; rj; jþ2Þ 1 1 1 1 q2 ¼ðxj xjþ1xjþ2xj xjþ2; rj; jþ1Þ 1 1 1 1 1 q3 ¼ðxj xjþ1xjþ2xj xjþ1; rjþ1; jþ2Þ 1 1 1 1 1 1 q4 ¼ðxj xjþ1xjþ2xj xjþ1xjþ2xjþ1; rj; jþ2Þ Braids and crossed modules 77
1 1 1 1 1 q5 ¼ðxj xjþ1xjþ2xjþ1; rjþ1; jþ2Þ 1 1 1 1 1 1 1 q6 ¼ðxj xjþ2xjþ1xjþ2xj xjþ1; rj; jþ1Þ 1 1 1 1 1 1 q7 ¼ðxj xjþ2xjþ1xjþ2xj xjþ1; rj; jþ2Þ 1 1 1 1 1 1 q8 ¼ðxj xjþ2xjþ1xjþ2xj xjþ1xjþ2; rj; jþ1Þ 1 1 1 1 1 1 q9 ¼ðxj xjþ2xjþ1xjþ2xj xjþ1xjþ2xjþ1xj; rjþ1; jþ2Þ 1 1 1 1 1 1 q10 ¼ðxj xjþ2xjþ1xjþ2xj ; rj; jþ2Þ 1 1 1 1 1 1 1 q11 ¼ðxj xjþ2xjþ1xj xjþ2xjþ1; rjþ1; jþ2Þ 1 1 1 1 1 q12 ¼ðxj xjþ2xjþ1xj xjþ1xjþ2; rj; jþ2Þ 1 1 1 1 1 q13 ¼ðxj xjþ2xjþ1xj ; rj; jþ1Þ 1 1 q14 ¼ðxjþ2xj ; rj; jþ2Þð6:3Þ
For B4, the construction stops at this stage. In particular, attaching a 3-cell to the geometric realization Kðx1; x2; x3; r1; 2; r2; 3; r1; 3Þ via the attaching map given by I1; 2; 3 yields a classifying space of B4. This amounts precisely to the operation of identifying faces of the truncated octahedron mentioned earlier. Thus the kernel 1 Z h i of p2ðK; K ÞAb ! ½B4 x1; x2; x3 is the free B4-module freely generated by ½I1; 2; 3 and the resulting B4-chain complex is a free resolution of the integers over B4; cf. e.g. [34]. Consequently
1 q 1 0 ! p2ðKÞ!p2ðK; K Þ!p1ðK Þ!B4 ! 1 ð6:4Þ is then a free crossed resolution of B4; cf. [18] for the notion of free crossed resolu- tion. When n d 5, additional identities among relations arise, according to the possible full subgraphs A2 A1 of An 1: For each pair ð j; kÞ with 1 c j < k c n 1 and j < k 2, the subgroup of Bn generated by xj, xjþ1, xk is isomorphic to the direct product B3 B2 of a copy of B3 with a copy of B2. Accordingly, consider the identity
Jj; jþ1; k ¼ðq1; q2; q3; q4; q5; q6; q7; q8Þð6:5Þ among relations arising from reading o¤ an identity sequence from a prism having base and top a hexagon labelled rj; jþ1 and having, furthermore, six lateral squares, each square being labelled with the relators rj; k or rjþ1; k:
1 1 1 1 q1 ¼ðxjþ1xj xjþ1; rj; jþ1Þ 1 1 1 1 1 q2 ¼ðxjþ1xj xjþ1xjxk ; rjþ1; kÞ 1 1 1 1 1 q3 ¼ðxjþ1xj xjþ1xk ; rj; k Þ 78 J. Huebschmann
1 1 1 1 q4 ¼ðxjþ1xj xjþ1xk ; rjþ1; kÞ 1 1 1 1 1 q5 ¼ðxjþ1xj xk xjþ1xjxjþ1; rj; k Þ 1 1 1 1 q6 ¼ðxjþ1xj xk xjþ1; rj; jþ1Þ 1 1 1 q7 ¼ðxjþ1xj xk ; rj; kÞ 1 1 q8 ¼ðxjþ1xk ; rjþ1; kÞ: ð6:6Þ
With the notation F3; 2 for the free group on xj, xjþ1, xk and C3; 2 for the free crossed F3; 2-module on the relators rj; jþ1, rj; k, rjþ1; k, the sequence
Z h i q 0 ! ½B3 B2 Jj; jþ1; k ! C3; 2 ! F3; 2 ! B3 B2 ! 1 ð6:7Þ is a free crossed resolution of B3 B2. Likewise, for each pair ð j; kÞ with 1 c j < k c n 1 and j < k 2, the subgroup of Bn generated by xj, xk 1, xk is isomorphic to the direct product B2 B3 of a copy of B2 with a copy of B3. Accordingly, consider the identity
Jj; k 1; k ¼ðq1; q2; q3; q4; q5; q6; q7; q8Þð6:8Þ among relations arising from reading o¤ an identity sequence from a prism having base and top a hexagon labelled rk 1; k and having, furthermore, six lateral squares, each square being labelled with the relators rj; k 1 or rj; k:
1 1 1 1 q1 ¼ðxk xk 1xk ; rk 1; kÞ 1 1 1 1 q2 ¼ðxk xk 1xk xk 1xj ; rj; kÞ 1 1 1 1 q3 ¼ðxk xk 1xk xj ; rj; k 1Þ 1 1 1 1 1 q4 ¼ðxk xk 1xk xj ; rj; k Þ ð6:9Þ 1 1 1 1 q5 ¼ðxk xk 1xj xk xk 1xk; rj; k 1Þ 1 1 1 1 q6 ¼ðxk xk 1xj xk ; rk 1; kÞ 1 1 1 1 q7 ¼ðxk xk 1xj ; rj; k 1Þ 1 1 1 q8 ¼ðxk xj ; rj; k Þ
When n d 6, more identities among relations arise, according to the possible full subgraphs A1 A1 A1 of An 1: For each triple ð j; k; lÞ with
1 c j < k < l c n 1; k j d 2; and l k d 2; Braids and crossed modules 79 the subgroup of Bn generated by xj, xk, xl is free abelian of rank three. The familiar commutator identity
½x; y y½x; z ½y; z z½y; x ½z; x x½z; y ¼e valid in any group entails the following identity
Jj; k; l ¼ðq1; q2; q3; q4; q5; q6Þ 1 1 1 ¼ððe; ri; jÞ; ðxj; ri; kÞðe; rj; kÞ; ðxk; ri; j Þðe; ri; k Þ; ðxi; rj; k ÞÞ ð6:10Þ among relations. This identity among relations also arises from reading o¤ an iden- tity sequence from a cube whose pairs of opposite lateral squares are labelled with the relators rj; k, rj; l,orrk; l. With the notation F2; 2; 2 for the free group on xj, xk, xl and C2; 2; 2 for the free crossed F2; 2; 2-module on the relators rj; k, rj; l, rk; l, the resulting sequence
Z 3 h i q 3 0 ! ½B2 Jj; k; l ! C2; 2; 2 ! F2; 2; 2 ! B2 ! 1 ð6:11Þ is then a free crossed resolution of B2 B2 B2. We can now prove Proposition 6.4. We will argue in terms of the cell decomposi- f tion of Pn=Sn made explicit in Proposition 6.2. When n d 4, as a Bn-module, p2ðKÞ is generated by the elements
A 1 ½Jj; k; l p2ðKÞ J p2ðK; K Þ; 1 c j < k < l c n 1:
Indeed, the classes of the identities written above as Jj; jþ1; jþ2 (1 c j c n 3) are killed by the 3-cells that occur in Statement (4) of Proposition 6.2; the classes of the identities Jj; jþ1; k and Jj; k 1; k (1 c j < k c n 1, j < k 2) are killed by the 3-cells that occur in Statement (5) of Proposition 6.2; and the classes of the identities
Jj; k; l; 1 c j < k < l c n 1; k j d 2; l k d 2; are killed by the 3-cells that occur in Statement (6) of Proposition 6.2. Since the f quotient complex Pn=Sn is aspherical, this argument establishes Proposition 6.4. 1 Alternatively, the induced map p2ðKÞ!p2ðK; K ÞAb is still injective and 1 p2ðK; K ÞAb amounts to the free Bn-module freely generated by the 2-cells of K or, equivalently, by the set R of relators. Let Z½Bn denote the integral group ring of Bn and, given a set Y, let Z½Bn hYi be the free Bn-module freely generated by Y. Let X f denote the set of Artin generators of Bn or, equivalently, that of 1-cells of Pn=Sn. The 1 Z h i 1 1 map p2ðK; K ÞAb ! ½Bn X induced by q : p2ðK; K Þ!p1ðK Þ has kernel the isomorphic image of p2ðKÞ and, together with the obvious map Z½Bn hXi ! Z½Bn , yields the beginning
q2 q1 Z½Bn hRi ! Z½Bn hXi ! Z½Bn 80 J. Huebschmann of the familiar free resolution of the integers in the category of Z½Bn -modules, well explored at various places in the literature [14], [16], [31], [34]. The kernel of q2 is canonically isomorphic to p2ðKÞ, and the next constituent
q3 q2 Z½Bn hJi ! Z½Bn hRi ! of the resolution kills the kernel p2ðKÞ of q2 in the sense that q3 maps onto p2ðKÞ, viewed as a Z½Bn -submodule of Z½Bn hRi; here the generating set J is that of the f f 3-cells of Pn=Sn, and the operator q3 is induced by the cellular boundary of Pn=Sn in dimension 3. These remarks establish Proposition 6.4 as well.
Remark 6.5. The identities among relations explored in the above proof arise from cell decompositions of the 2-sphere which, in turn, rely on the poset geometry discussed in Subsection 6.2 above. The cell decomposition of the 2-sphere dual to the permutahe- dron P4 is a 14-gon and the cell decomposition of the 2-sphere dual to the prism having base and top a hexagon and having, furthermore, six lateral squares, is an octagon. The 14-gons and octagons are related with the geometry of posets of maximal chains of the symmetric group on n letters discussed in combinatorics, cf. e.g. [22].
7 Universal central extensions The Schur multiplicator of a group (later identified as the second homology group) was originally discovered by Schur in the search for central extensions to realize projective representations by linear ones. In this spirit, we will now push further the situation of Example 3.5. The starting point is the (familiar) observation that, given the group G with H1ðGÞ free abelian or zero, for any abelian group A, the universal coe‰cient map 2 H ðG; AÞ!HomðH2ðGÞ; AÞð7:1Þ is an isomorphism whence the central extensions of G by the abelian group A are then classified by HomðH2ðGÞ; AÞ. We will refer to a central extension
0 ! H2ðGÞ!U ! G ! 1 ð7:2Þ as being universal provided it has the following universal property: Given the central extension p e : 0 ! A ! E ! G ! 1; ð7:3Þ the identity of G lifts to a commutative diagram
0 ! H2?ðGÞ ! U? ! G? ! 1 ? ? ? ? ? ? yQ yY yId ð7:4Þ p 0 ! A ! E ! G ! 1 Braids and crossed modules 81 of central extensions in such a way that the assignment to e of Q induces the universal coe‰cient map (7.1). When H1ðGÞ is free abelian or zero (that is, G is perfect), it is straightforward to construct a universal central extension of G and standard abstract nonsense reveals that a universal extension is then unique up to isomorphism. Here the notion of universal central extension is to be interpreted with a grain of salt, though, since this notion is lingua franca only for the case where G is perfect [26] and can then be characterized in other ways.
Theorem 7.1. Under the circumstances of Theorem 4.1, the central extension (4.1) is the universal central extension of G.
Proof. This is a consequence of the freeness of the G-crossed module C: Given a central extension e of the kind (7.3), lift the injection S J G to an injection S ! E; keeping in mind that, from the central extension, E acquires a G-crossed module structure, let Ye : C ! E be the resulting morphism of G-crossed modules and let Qe : H2ðGÞ!A denote the restriction of Ye to the kernel H2ðGÞ. The central exten- sion e represents a class ½e A H2ðG; AÞ and, furthermore, the assignment to ½e of Qe A HomðH2ðGÞ; AÞ is a conceptual description of the universal coe‰cient map 2 H ðG; AÞ!HomðH2ðGÞ; AÞ. r
Thus, in particular, the free Bn-crossed module Cn in Theorem 5.2 above yields the universal central extension of Bn. Likewise the central extension (4.4) is the universal central extension of the free abelian group A on the set S. Furthermore, the map (5.15) is the universal map induced from the injection of A in B4 (or more generally Bn) via the universal property of the universal central extension.
Acknowledgements. I am indebted to J. Stashe¤ for a number of comments on a draft of the paper, to F. Cohen for discussion about the generating cycle of H2ðBnÞ, see Remark 5.3, to C. Berger for discussion about the combinatorics of the permuta- hedron and the operadic consequences thereof, and to the referee for suggesting a number of improvements of the exposition.
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Received 15 June, 2010; revised 26 January, 2011 J. Huebschmann, USTL, UFR de Mathe´matiques, CNRS-UMR 8524, 59655 Villeneuve d’Ascq Cedex, France E-mail: [email protected]