ARTIN BRAID GROUPS AND HOMOTOPY GROUPS

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1 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS JINGYAN LI AND JIE WU Abstract. We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a summand of the center of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. These results give some deep and fundamental connections between the braid groups and the general higher homotopy groups of spheres. 1. Introduction In this paper, we investigate the Brunnian subgroups of the Artin braid groups and its connections with homotopy theory. The crucial point is that these subgroups of the braid groups have deep connections with the general homotopy groups of the sphere. From establishing these relations, one obtains various braided means of the general homotopy groups of the sphere. A (geometric) braid is called Brunnian if (1) it is a pure braid and (2) it becomes a trivial braid by removing any of its strands. Since the composition of any two Brunnian braids is still Brunnian, the set of Brunnian braids is a (normal) subgroup of the braid group B n which is denoted by Brun n. As a group, Brun n is a free group of infinite rank with a set of generators to be described in [2, 18, 34] for n 3. However as a normal subgroup of the pure braid group P n, Brun n has finitely many normal generators described as follows. Recall that B n is generated by subject to the braiding relations: σ 1, σ 2,..., σ n 1 (1). σ i σ j = σ j σ i for i j 2, and (2). σ i σ i+1 σ i = σ i+1 σ i σ i+1 for each i. The symmetric group S n is the quotient group of B n subject to the following additional relations: (3) σi 2 = 1 for each i. The pure braid group P n is defined to be the kernel of the quotient map B n S n, with a set of generators given by A i,j = σ j 1 σ j 2 σ i+1 σ 2 i σ 1 i+1 σ 1 j 2 σ 1 j 1 Partiall supported by the Academic Research Fund from Shijiazhuang Railway Institute. Partially supported by the Academic Research Fund from the National University of Singapore. 1

2 2 JINGYAN LI AND JIE WU for 1 i < j n. By Theorem 3.7, Brun n is the normal subgroup of P n generated by the iterated commutators [[[A 1,2, A i2,3], A i3,4],..., A in 1,n] for 1 i t t and 2 t n 1. Thus the quotient groups P n /Brun n and B n /Brun n are finitely presented. It is a canonical way to describe Brun n as the intersection of the subgroups of P n. Let d i : B n B n 1 be the function obtained from removing the ith strand of the braids for 1 i n. Then, restricted to pure braids, d i : P n P n 1 is a group homomorphism. (Note. The function d i : B n B n 1 is not a homomorphism but it satisfies the crossed condition described in Section 2.) From the definition, Brun n = n i=1 Ker(d i : P n P n 1 ) and so the homomorphisms d i : P n P n 1 induces a faithful representation n P n /Brun n i=1 P n 1 that is the group P n /Brun n can be regarded as a subgroup of a product of P n 1. In particular, P n /Brun n is torsion free. For introducing the notion of boundary Brunnian braids, there is a canonical automorphism θ of P n described as follows. Let F (C, n) = {(z 1,..., z n ) z i z j for i j} be the configuration space over the plane. Let φ: F (C, n) F (C, n) be the map defined by ( ) φ(z 1, z 2,..., z n ) = 0,,,..., z 2 z 1 z 3 z 1 z n z 1 corresponding geometrically to the reflection map in C about the unit circle centered at z 1. Then φ induces an isomorphism φ : P n = π 1 (F (C, n)) P n = π 1 (F (C, n)) under a choice of the basepoint for F (C, n). Notice that there is a homomorphism χ : B n B n that sends each standard generator σ i to its inverse σ 1 i, because such a homomorphism preserves the relations for the braid group. In pictures, χ(β) is the mirror reflection of the braid β. Let θ = χ φ. Namely θ is the product of two natural reflections χ and φ. By Lemma 2.3, the action of θ on P n is given by θ(a i,j ) = A i,j for 2 i < j n, and θ(a 1,j ) = A 1 1,j A 0,jA 1,j, where A 0,j = (A j,j+1 A j,j+2 A j,n ) 1 (A 1,j A j 1,j ) 1 = (σ j σ j+1 σ n 2 σ 2 n 1σ n 2 σ j ) 1 (σ j 1 σ 2 σ 2 1σ 1 σ j 1 ) 1 is an element in P n for 1 j n. According to Corollary 3.4, there is an action of B n+1 on P n, where the action of σ 1 B n+1 on P n is given by θ and the action of σ i B n+1 (2 i n) on P n is given by the conjugation action of σ i 1 B n on P n. Thus the automorphism θ plays an important role for B n+1 -action on P n. By restricting to the Brunnian braids, Brun n is not invariant under the action of θ but has the property that d i θ(β) = 1 and d 1 θ(β) Brun n 1 for β Brun n and i 2. Namely for given a Brunnian braid β θ(β) becomes a trivial braid after removing any of its strands except the first. A braid of the form d 1 θ(β) for some

3 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 3 β Brun n is called an (n 1)-strand boundary Brunnian braid. The set of (n 1)- strand boundary Brunnian braids forms a subgroup of Brun n 1, denoted by Bd n 1, that is Bd n 1 = d 1 θ(brun n ). In pictures, the boundary Brunnian braids can be described as follows. Let D 2 (r) be the disk centered at 0 with radius r sufficiently large. The geometric n-strand pure braids are given inside the cylinder D 2 (r) I with the first strand given as the straight line segment in the cylinder D 2 (r) I from the origin of the top disk D 2 (r) {1} down to the bottom disk D 2 (r) {0}, and the rest of the strands starting from the (distinct) points q 2,..., q n ordered clockwise lying in the first quadrant of the unit circle S 1 inside D 2 (r) {1} down to the points q 2,..., q n in D 2 (r) {0}, respectively. Then the (n 1) boundary Brunnian braids are obtained from an n-strand Brunnian braid β by applying the reflection about S 1 I to the strands of β except the first strand followed by the mirror reflection and then removing the first strand of the resulting braid. The group Bd n has a connection with mapping class groups. Let Γ 0,n be the mapping class group of n-punctured sphere. Consider B n as the mapping class group of n-punctured disk. The canonical embedding of the disk into the sphere (as northern hemisphere) induces a group homomorphism q : B n Γ 0,n. By Corollary 3.14, the group Bd n is the commutator subgroup in B n of Brun n and the kernel of q. From Proposition 3.15, Bd n is the normal subgroup of P n generated by the iterated commutators [[[A 1,2, A i2,3], A i3,4,..., A in 1,n], A 0,in ] for 1 i t t and 2 t n. Thus the groups P n /Bd n and B n /Bd n are finitely presented. Two fundamental questions then arise naturally: 1) Is the group P n /Bd n torsion free? 2) What is the center of B n /Bd n? The second question is a special case of the conjugation problem on braids. Namely, how to determine a braid β B n such that the conjugation σ i βσ 1 i lies in the coset βbd n for each 1 i n 1. For a subgroup H of G, let (H, G) = {x G x q H for some q Z} denote the set of the roots of H in G. Then (Bd n, P n )/Bd n is the set of torsion elements in P n /Bd n. Surprisingly the answers to the above questions can be given in terms of the homotopy group π n (S 2 ). Denote by Z(G) the center of a group G. Theorem 1. Let n 4. 1) (Bd n, P n ) = Brun n θ(brun n ) with an isomorphism of groups (Bdn, P n )/Bd n = πn (S 2 ). 2) There are isomorphisms of groups Z(P n /Bd n ) = π n (S 2 ) Z and Z(B n /Bd n ) = {α π n (S 2 ) 2α = 0} Z. By moving our steps to the next, consider the mirror reflection χ: B n B n. Given a subgroup G of B n, one may ask what are the mirror symmetric braids β subject to G, that is, the braids β satisfying the equation of cosets χ(β)g = βg. If G is the trivial subgroup, it is well-known that the mirror reflection χ on B n is free and so the trivial braid is the only mirror symmetric braid subject to the trivial subgroup G = {1}. For general cases that G is a non-trivial subgroup of

4 4 JINGYAN LI AND JIE WU B n, the question on mirror symmetric braids becomes very nontrivial. In the case G = Bd n, the answer is again given in term of the homotopy group π n (S 2 ). Let Fix φ (G) denote the subgroup of the fixed-points of an action φ on a group G. Theorem 2. The subgroup Bd n is invariant under the mirror reflection χ. Moreover there is an isomorphism of groups for n 3. Fix χ (B n /Bd n ) = π n (S 2 ) The methods for proving Theorems 1 and 2 are described as follows. There is a construction of almost simplicial group P = {P n } n 0 that can be regarded as a model for S 2, where the face operations d i : P n P n 1 is given by removing the ith strand for 1 i n with d 0 = d 1 θ and degeneracy operations s i : P n P n+1 is given by doubling the ith strand for 1 i n. It should be pointed out that P is not a simplicial group as 0th degeneracy is missing in the structure. However many simplicial techniques can still be applied to P. In particular, it will be directly proved that Z(P n /Bd n ) = π n (S 2 ) using the almost simplicial structure on P. The determination of the center of B n /Bd n arises the systematic relations between the conjugation actions of B n on P n and the face operations in P. The proof of Theorem 2 arises from the fact that the mirror reflection χ on P is a morphism of almost simplicial group. By using the simplicial arguments, the mirror symmetric braids in B n /Bd n are given by the elements in the homotopy group. On the other hand, the Moore looping of χ induces a continuous map Ω[ 1]: ΩS 2 ΩS 2 which then induces the identity map on homotopy groups π n (S 2 ) for n 3 by the classical methods in homotopy theory using Hopf invariants. Thus the methods in this article is a kind of combination of the techniques in homotopy theory and group theory, where the simplicial model provides a bridge for two different areas. Some historical remarks concerning the homotopy groups and this paper are given next. The fundamental group owes its existence to Poincaré [26]. Čech [7] suggested how to define higher homotopy groups in 1932 without pursuing the notion, and it was Hurewicz [16] who first studied them in It was originally conjectured that the homotopy groups of spheres are isomorphic to their homology groups. Then Hopf invented the Hopf map [15]. The determination of higher homotopy groups of spheres became the fundamental problem in homotopy theory from then on. Although the determination of the general homotopy groups is beyond current technology, much progress has been made over time, for instance [9, 17, 30]. The computation of the homotopy groups of spheres can be found in [13, 31] and other references, where the homotopy groups π n (S 2 ) are known for n 64 according to [13]. Up to this range, by using Theorems 1 and 2, we are able to determine the centers of P n /Bd n and B n /Bd n, as well as the mirror symmetric braids in B n /Bd n. For instance, π 4 (S 2 ) = π 5 (S 2 ) = Z/2. Thus Z(P n /Bd n ) = Z(B n /Bd n ) = Z/2 Z for n = 4, 5. By using the fact that π 6 (S 2 ) = Z/12, Z(P 6 /Bd 6 ) = Z/12 Z but Z(B 6 /Bd 6 ) = Z/2 Z. Some important properties on general homotopy groups π n (S 2 ) are as follows. It was known by Serre [30] that π n (S 2 ) is a finite abelian group for n 4. Let Tor p (G) denote the p-torsion component of an abelian group G for a prime integer p. It was proved by James [17] that 4 Tor 2 (π n (S 2 )) = 0.

5 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 5 Namely Tor 2 (π n (S 2 )) is a finite direct sum of Z/2 and Z/4. Furthermore, a result of Selick [28] states that p Tor p (π n (S 2 )) = 0 for the prime p > 2, namely Tor p (π n (S 2 )) is a finite direct sum of Z/p for p > 2. Thus, for n 4, the group π n (S 2 ) is determined by the orders of the groups (1) π n (S 2 ) = Fix χ (B n /Bd n ) = (Bd n, P n )/Bd n = Tor(Z(P n /Bd n )) and (2) {α π n (S 2 ) 2α = 0} = Tor(Z(B n /Bd n )). As a combinatorial tool for studying homotopy theory, simplicial groups were first studied by J. C. Moore [24]. The classical Moore theorem states that π ( G ) = H (NG), where G is the geometric realization of G and NG is the Moore chain complex of G described in Section 2. Milnor [22] then proved that any loop space is (weakly) homotopy equivalent to a geometric realization of a simplicial group, and so, theoretically speaking, the homotopy groups of any space can be determined as the homology of a Moore chain complex. It is possible that two simplicial groups with the same homotopy type have sharply different group structures. Simplicial group models for loop spaces have been studied by many people, see for instance [1, 6, 8, 19, 23, 24, 27, 32, 33]. Different simplicial group models for the same loop space may give different homotopy information. For example, the classical Adams spectral sequence arises as the associated graded by taking the mod p descending central series of Kan s G-construction on reduced simplicial sets, [4, 5, 12]. On the other hand, one could have a perfect simplicial group model (that is, the abelianization is the trivial group) for certain loop spaces by using Carlsson s construction [33]. For this model, the descending central series will not give any information as the groups are perfect, but word filtration provides different information. By using Milnor s F [K]-construction on the simplicial circle, a combinatorial description of the general homotopy groups π n (S 2 ) was obtained in [34], where it was proved that the general homotopy group π n (S 2 ) is isomorphic to the center of a combinatorially given group G n with n generators and certain systematic (infinitely) relations [34, Theorem 1.4]. It was then asked by many people whether there is a finitely presented group whose center is given by π n (S 2 ). Theorem 1 in this article gives a positive answer to this question. A connection between the braid groups and the general homotopy groups of S 2 was found in [35], where it was proved that the Artin braid group B n acts on the group G n and the homotopy group π n (S 2 ) is given by the fixed set of the pure braid group P n action on G n [35, Theorem 1.2]. Moveover it was proved in [2, Theorem 1.2] that π n (S 2 ) are given by the (n + 1)-strand Brunnian braids over the sphere modulo the (n + 1)-strand Brunnian braids over the disk for n 4. Some relations between the homotopy groups and Vassiliev invariants have been studied in [10, 11]. Theorems 1 and 2 in this article then give further connections between the braid groups and the homotopy groups for addressing the conjugation problem and the mirror reflection problem on the braids. It should be pointed out that the simplicial group constructions G using free groups or free products arise computational difficulties for determining the homology of the Moore chain complex NG because the Moore chains are given by the intersections of the subgroups of G. Theorem 2.11 in this article states that the

6 6 JINGYAN LI AND JIE WU homotopy groups π n (S 2 ) occur as the summands of the homology of the chain complex given by abelianization Brun ab n of the Brunnian braid group Brun n with differential induced by d 1 θ. This result arises a chance to represent the homotopy groups of spaces as the homology of abelian chain complexes. Notice that the conjugation action of B n on Brun n induces a B n -module structure on Brun ab n. By Theorem 3.7, as a B n -module, Brun ab n is finitely generated. The B n -module Brun ab n has the connections with the important symmetric group module Lie(n) studied in [8, 29, 36, 37, 38] that plays a key role for studying some long-standing problems in homotopy theory and the modular representation theory of Lie modules over the general linear groups. The article is organized as follows. In Section 2, we investigate the simplicial and -structures on braids. The conjugation action of B n on the Brunnian braids and the boundary Brunnian braids are given in Section 3. In Section 4, we give the proofs of Theorems 1 and Simplicial and -structure on braids 2.1. The crossed simplicial group B. Let O be the category of finite ordered sets and ordered functions, where a function f is ordered if f(x) f(y) when x y. The category O has objects [n] = {0,..., n} for n 0 and morphisms are generated by the coface functions d i : [n 1] [n] (which misses i) and the codegeneracy functions s i : [n + 1] [n] (which hits i twice) for 0 i n. Recall that a simplicial object X over a category C is a contravariant functor from O to C. In other words, X = {X n } n 0, where X n = X ([n]). The face d i : X n X n 1 is given by d i = X (d i ) and the degeneracy s i : X n X n+1 is given by s i = X (s i ) for 0 i n. The simplicial identities follow from the well-known formulas for functions d i and s j in the category O. A simplicial object over sets (resp. monoids, groups, Lie algebras, spaces, etc) is called a simplicial set (resp. monoid, group, Lie algebra, space, etc). Standard references for the theory of simplicial objects are [12, 21]. The standard simplicial structure on the sequence of symmetric groups is as follows. Example 2.1. Let S n+1 denote the symmetric group of bijections of the symbols 0, 1,..., n. Sometimes right actions of S n are used by requiring i σ = σ 1 (i). Let S = {S n+1 } n 0 be the sequence of symmetric groups of degree n + 1. Then S is a simplicial set in the following way. The face d i : S n+1 S n is uniquely determined by the commutative diagram [n 1] d i σ [n] (2.1) d i (σ) [n 1] d i σ [n] for any σ S n+1, that is, d i (σ) = s i σ d σ 1 (i). The degeneracies s i : S n+1 S n+2 are determined uniquely by requiring (2.2) s i (σ)(σ 1 (i)) = i, s i (σ)(σ 1 (i) + 1) = i + 1

7 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 7 and the diagram [n + 1] s i σ [n] (2.3) s i (σ) [n + 1] s i σ [n] commutes, that is (s i ) 1 σs σ 1 (i) (k) k σ 1 (i), σ 1 (i) + 1, s i (σ)(k) = i k = σ 1 (i), i + 1 k = σ 1 (i) + 1. Note that Equation 2.2 follows from the commutative diagrams 2.1 and 2.3 together with simplicial identities: s 1 s 0 = s 0 s 0 for the case n = 0 and the expression for d k s i, k i, i + 1, for the case n 1. A crossed simplicial group is a simplicial set G = {G n } n 0 for which each G n is a group, together with a group homomorphism µ: G n S n+1, g µ g for each n, such that (i) µ is a simplicial map, and (ii) for 0 i n, d i (gg ) = d i (g)d i µg (g ) and s i (gg ) = s i (g)s i µg (g ). An important example is that the sequence of Artin braid groups B = {B n+1 } n 0 is a crossed simplicial group with the faces and degeneracies described as follows: Given an (n + 1)-strand braid β B n+1, d i β is obtained by removing (i+1) st strand braid and s i β is obtained by doubling (i+1)-strand for 0 i n, where the strands are counted from initial points and µ: B n+1 S n+1 be the canonical quotient. Since the restriction of d i and s i to the pure braid groups are group homomorphisms, the sequence of groups P = {P n+1 } is a simplicial group. (See [2, Theorem ] for crossed simplicial arising from configuration spaces.) Let σ 1,..., σ n be the standard generators for B n+1 and let (2.4) A i,j = σ j 1 σ j 2 σ i+1 σ 2 i σ 1 i+1 σ 1 j 2 σ 1 j 1 be the generators for P n+1 for 1 i < j n. The braids σ i and A i,j are pictured as follows, where the geometric braids are from top to bottom according to [3].

8 8 JINGYAN LI AND JIE WU 1 i i+1 n i j σ i A i,j By drawing the pictures of braids, the faces and degeneracies on the generators for B n+1 and P n+1 are as follows: σ i 1 if k + 1 < i d k σ i = 1 if k + 1 = i, i + 1 σ i if k + 1 > i + 1, (2.5) s k σ i = d k A i,j = s k A i,j = σ i+1 if k + 1 < i σ i+1 σ i if k + 1 = i σ i σ i+1 if k + 1 = i + 1 σ i if k + 1 > i + 1, A i 1,j 1 if k + 1 < i 1 if k + 1 = i, j A i,j 1 if i < k + 1 < j A i,j if k + 1 > j, A i+1,j+1 if k + 1 < i A i,j+1 A i+1,j+1 if k + 1 = i A i,j+1 if i < k + 1 < j A i,j A i,j+1 if k + 1 = j A i,j if k + 1 > j. Recall that the mirror reflection χ: B n B n is the group homomorphism such that χ(σ i ) = σ 1 i. Proposition 2.1. The mirror reflection χ on braids induces a simplicial map χ: B B, that is d i χ = χ d i and s i χ = χ s i for each i. Proof. Let µ: B n S n be the canonical quotient. Note that µ(σi 2 ) = 1. The induced action of χ on S n is trivial. From the crossed identities d i (xx ) = d i x d i µ(x) x and s i (xx ) = s i x s i µ(x) x together with the fact that µ(χ(x)) = µ(x), it suffices to show that χ d i (σ j ) = d i χ(σ j ) and χ s i (σ j ) = s i χ(σ j ) for any i and j, while these identities hold by a direct computation using the formulae in Equation The simplicial group model for ΩS 2. Let X be a pointed simplicial set. Let X 0 be the basepoint. The basepoint in X n is s n 0. Let F [X ] n be the free group generated by X n subject to the single relation that s n 0 = 1. (Note. By the simplicial identities, s n 0 = s in s in 1 s i1 for any sequence (i 1, i 2,..., i n ) with 0 i k k 1.) Then we obtain the simplicial group F [X ] = {F [X ] n } n 0 with the faces and the degeneracies induced by those of X. The simplicial group F [X ]

9 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 9 is called Milnor s free group construction on X. construction of F [X ] is as follows. An important property of the Theorem 2.2. Let X be any pointed simplicial set. Then the geometric realization F [X ] of F [X ] is homotopy equivalent to ΩΣ X. Note. The geometric realization of F [X ] is the group completion of the James construction on X. In Milnor s original article [23], the above theorem is stated in case X is a reduced simplicial set. For the general case, one can see for instance [33]. Now let S 1 be the simplicial 1-sphere. The elements in Sn 1 can be listed as follows. S0 1 = { }, S1 1 = {s 0, σ}, S2 1 = {s 2 0, s 0 σ, s 1 σ}, S3 1 = {s 3 0, s 2 s 1 σ, s 2 s 0 σ, s 1 s 0 σ}, and in general Sn+1 1 = {s n+1 0, x 0,..., x n }, where x j = s n ŝ j s 0 σ. The faces d i : Sn+1 1 Sn 1 and degeneracies s i : Sn+1 1 Sn+2 1 are given by the formulae (2.6) d i x j = d i s n ŝ j s 0 σ = s i x j = s i s n ŝ j s 0 σ = s n 0 if j = i = 0 or i = j + 1 = n + 1, x j if j < i, x j 1 if j i, { xj if j < i, x j+1 if j i. Consider the special case of Milnor s free group construction for S 1, F [S 1 ]. According to Theorem 2.2, F [S 1 ] is a simplicial group model for ΩS 2. As a sequence of groups, the group F [S 1 ] n is the free group of rank n generated by x 0, x 1,..., x n 1 with faces as above. Tietze transformations may be used to change the free generators of the free group F [S 1 ] n+1, so as to reformulate the faces d i in a canonical way. Let y 0 = x 0 x 1 1,..., y n 1 = x n 1 x 1 n and y n = x n in F [S 1 ] n+1. Clearly {y 0, y 1,..., y n } is a set of free generators for F [S 1 ] n+1 with given by (2.7) d k y j and s k y j (0 k n + 1, 0 j n) d k y j = d k (x j x 1 j+1 ) = s k y j = y j 1 if k < j if k = j + 1 y j if k > j + 1, y j+1 if k < j + 1 y j y j+1 if k = j + 1 y j if k > j + 1, where y 1 = (y 0 y 1 y n 1 ) 1 and in this formula x n+1 = 1. Under the generating system of y j s, the faces d i with i > 0 are projection maps in the sense that d i sends y i 1 to 1 and other generators to the generators for F [S 1 ] n so as to retain the order. The first face d 0 differs from the others as d 0 sends y 0 to the product element (y 0 y 1 y n 1 ) 1 and each other generator y j to y j 1 for F [S 1 ] n The -group P. A sequence of sets X = {X n } n 0 is called a -set if there are functions d i : X n X n 1 for 0 i n such that d j d i = d i d j+1 for i j. f = {f n } n 0 : X X is called a -map if d i f n = f n 1 d i. Moreover a -group G = {G n } n 0 consists of a -set G for which each G n is a group and each d i is

10 A 0,j 10 JINGYAN LI AND JIE WU a group homomorphism. The Moore complex NG = {N n G} n 0 of a -group G is defined by n (2.8) N n G = Ker(d i : G n G n 1 ). i=1 For a -group G, it is straightforward to check that d 0 (N n G) N n 1 G and so NG with d 0 is a chain complex of groups. An element in (2.9) Bd n G = d 0 (N n+1 G) is called a Moore boundary and an element in (2.10) Z n G = Ker(d 0 : N n G N n 1 G) is called a Moore cycle. The nth homotopy π n (G) is defined to be the coset (2.11) π n (G) = H n (NG) = Z n G/Bd n G. We are interested in a particular -group arising from braids. Let (2.12) A 0,j = (A j,j+1 A j,j+2 A j,n ) 1 (A 1,j A j 1,j ) 1 = (σ j σ n 2 σn 1σ 2 n 2 σ j ) 1 (σ j 1 σ 2 σ1σ 2 2 σ j 1 ) 1 in P n for 1 j n. The picture of A 0,j is given as follows: j-1 j 1 j+1 n According to [2, Lemma 6.5.2], there is a group homomorphism : P n P n 1 for each n such that (2.13) (A i,j ) = A i 1,j 1 for 1 i < j n. (Note. A 0,j was denoted by A 1,j in [2].) Lemma 2.3. In the simplicial group P, the homomorphism : P n P n 1 has the following properties: (1). (A 0,j ) = 1 for 1 j n. (2). d i = d i+1 for 0 i n 2. (3). = d 0. (4). χ n 1 = χ n, where χ n : P n P n is the mirror reflection. (5). s i = s i 1 for i 1. (6). There is an automorphism θ : P n P n such that { Ai,j if 2 i < j n θ n (A i,j ) = A 1 1,j A 0,jA 1,j if i = 1 < j n. Moreover = d 0 θ. Proof. Assertions (1)-(3) are given in [2, Lemma 6.5.2]. Assertion (4) follows from the lines in the proof of [2, Lemma 6.5.2] by drawing the pictures of the braids for χ n 1 (A i,j ) and χ n (A i,j ), respectively.

11 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 11 (5). Note that s k A i,j = for k 1 and 1 i < j, and A i+1,j+1 = A i,j if k + 1 < i, (A i,j+1 A i+1,j+1 ) = A i 1,j A i,j if k + 1 = i, A i,j+1 = A i 1,j if i < k + 1 < j, (A i,j A i,j+1 ) = A i 1,j 1 A i 1,j if k + 1 = j, A i,j = A i 1,j 1 if k + 1 > j s k 1 A i,j = s k 1 A i 1,j 1 = for k 1 and 2 i < j. Thus A i,j if k + 1 < i, A i 1,j A i,j if k + 1 = i, A i 1,j if i < k + 1 < j, A i 1,j 1 A i 1,j if k + 1 = j, A i 1,j 1 if k + 1 > j s k A i,j = s k 1 A i,j for 2 i < j. For the case that i = 1 < j, from the above computation, we have A 0,j if 1 < k + 1 < j, s k A 1,j = A 0,j 1 A 0,j if k + 1 = j, A 0,j 1 if k + 1 > j. Since s k 1 A 0,j 1 is the braid given by doubling the kth strand of the braid A 0,j 1, we have s k 1 A 0,j 1 = s k A 1,j by drawing the picture of the braids. Thus s k 1 A 1,j = s k 1 A 0,j 1 = s k A 1,j and so s k A i,j = s k 1 A i,j for any 1 i < j n. (6). Let F (C, n) = {(z 1,..., z n ) C n z i z j for i j} be the configuration space over the complex plane. Let φ: F (C, n) F (C, n) be the map defined by φ(z 1, z 2,..., z n ) = ( 0, 1 z 2 z 1, 1 z 3 z 1,..., 1 z n z 1 corresponding geometrically (coordinate-wise) to inversion in C with respect to the unit circle centered at z 0 with 0 in the first coordinate. First we show that φ is a homotopy equivalence. By Fadell-Neuwirth Theorem [14], the coordinate projection π : F (C, n) C (z 1,..., z n ) z 1 is a fibre bundle. Since C is contractible, the inclusion i: F (C {0}, n 1) F (C, n) (z 2,..., z n ) (0, z 2,..., z n ) is a homotopy equivalence. Observe that φ maps into the fibre F (C {0}, n 1). Since φ i, as a self map of F (C {0}, n 1), is a homeomorphism, φ is a homotopy equivalence. Thus φ induces an isomorphism φ : P n = π 1 (F (C, n)) = P n = π 1 (F (C, n)). Choose the basepoint (q 1, q 2,..., q n ) F (C, n) by setting q 1 = 0 and letting q 2, q 3,..., q n be the points, ordered clockwise, lying in the first quadrant of the unit circle. By using the correspondence between the braids and the paths in the configuration space F (C, n), the figure A i,j in [3, Figure 4, p.21] corresponds a loop λ(t) = (λ 1 (t), λ 2 (t),..., λ n (t)), )

12 12 JINGYAN LI AND JIE WU where λ s (t) = q s is the constant path for s j as it corresponds to the straight line and the jth component λ j (t) is pictured as follows: i j j 1 1 A 1,j A i,j i>1 The correspondent braid of the loop φ λ(t) = (0, q 2,..., q j 1, 1 λ j (t), q j+1,..., q n ) is the mirror reflection χ(a i,j ) of A i,j if i > 2 from the above picture by taking inversion of λ j (t) with respect to the unit circle. For the case that i = 1, the correspondent braid of φ λ(t) is pictured as follows : 1 j 1 j = φ(a 1,j ) Thus { χ(ai,j ) if i > 1 φ (A i,j ) = χ(a 1 1,j )χ(a 0,j)χ(A 1,j ) if i = 1. Define θ = χ φ. Then θ is an automorphism of P n such θ(a i,j ) = A i,j for i > 1 and θ(a 1,j ) = A 1 1,j A 0,jA 1,j. Clearly d 0 θ n = and hence the result. From Assertions (1)-(3) of the above lemma, the simplicial group P has the additional face which can be treated as the new 0-th face. Assertion (5) gives the relations between new 0-th face and the existing degeneracies of P. From these information, we construct the new sequence of pure braid groups P = {P n } n 0, where P 1 = P 0 = {1}. The faces d P i is given by d P 0 = and d P i = dp i 1 for i > 0. In other words, in P, d 0 is given by : P n P n 1, and d i : P n P n 1 is given by deleting the ith strand of the braids. The degeneracies s P i = sp i 1 for i > 0, namely

13 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 13 the ith degeneracy in P is given by doubling the ith strand of the braids. Here 0th degeneracy in P is not defined. Motivated from this example, a -group G = {G n } n 0 is called an almostsimplicial group if there exists homomorphism s i : G n G n+1 for 1 i n such that simplicial identities hold except for the cases where s 0 is involved in the identities. Let G = {G n } n 0 and G = {G n} n 0 be almost simplicial groups. A morphism of almost simplicial groups means a -homomorphism f = {f n } n 0 : G G such that s i f n = f n+1 s i for i 1 and n 0. By Lemma 2.3 and Proposition 2.1, we have the following: Proposition 2.4. The sequence of groups P = {P n } n 0 is an almost simplicial group under the faces d i = d P i for i 0 and the degeneracies s i = s P i for i 1. Moreover the mirror reflection χ on braids induces a morphism of almost simplicial groups χ: P P, and d 0 = d 1 θ n : P n P n 1. For an almost-simplicial group G, the Moore path P G of G is defined by ( P G) n = {x G n+1 d n+1 0 (x) = 1} with d P j : ( P G) n ( P G) n 1, 0 j n, and s P i : ( P G) n ( P G) n+1, 1 i n, where d P j (x) = d j (x) and s P i (x) = s i (x). The map p: P G G is defined by p: ( P d n+1 G) n G n+1 G n. Since d j d n+1 = d n d j for 0 j n and d n+2 s i = s i d n+1 for 1 i n, the map p is a morphism of almost-simplicial -group. The Moore loop ΩG of G is defined to be the kernel of p. In other words, ( ΩG) n = {x G n+1 d n+1 0 (x) = 1 and d n+1 (x) = 1} and each face d Ω j = d j and s Ω j = s j. Proposition 2.5. There is an isomorphism of almost-simplicial groups F [S 1 ] = ΩP. In particular, π n (P) = π n 1 (F [S 1 ]) = π n 1 (ΩS 2 ) = π n (S 2 ). Proof. Note that the iterated faces d n+1 0 : P n+1 P 0 = 1 is the trivial map. The last face d n+1 : P n+1 P n is given by deleting the last strand. Thus the group ( ΩP) n = Ker(d n+1 ) is freely generated by A i,n+1 for 1 i n. From Equations 2.5 and 2.13, we have d k A i,n+1 = s k A i,n+1 = A i 1,n if k < i, 1 if k = i, A i,n if i < k < n + 1, A i+1,n+2 if 1 k < i, A i,n+2 A i+1,n+2 if k = i, A i,n+2 if i < k < n + 1. By comparing with Equation 2.7, the homomorphism F [S 1 ] n ( ΩP) n, y i A i+1,n+1 for 0 i n 1 induces an isomorphism of almost-simplicial groups.

14 14 JINGYAN LI AND JIE WU Proposition 2.6. There does not exist a sequence of functions s 0 : P n P n+1 such that P = {P n } is a simplicial set with s 0 and the existing faces and degeneracies of P. Proof. Suppose that the 0-th degeneracy function s 0 : P n P n+1 exists. From the simplicial identity s 0 d 0 = d 0 s 1, there is a commutative diagram P n+1 d 0 Pn s 1 s 0 d0 Pn+1. P n+2 Since d 0 : P n+1 P n is an epimorphism and the composite s 0 d 0 = d 0 s 1 is a group homomorphism, the function s 0 : P n P n+1 is a group homomorphism. It follows that P is a simplicial group. Note that ( ΩP) n is a free group of rank n. By [35, Corollary 2.16], ΩP is isomorphic to F [S 1 ] as a simplicial group and so the geometric realization P S 2. Recall that the geometric realization of a simplicial group is a topological group. We obtain a contradiction to the fact that S 2 is not an H-space and hence the result. Let X be a -set. The elements x 0,..., x i 1, x i+1,..., x n X n 1 are called matching faces with respect to i if d j x k = d k x j+1 for k j and k, j +1 i. Recall that a -set X is fibrant if it satisfies the following homotopy extension condition for each i: Let x 0,..., x i 1, x i+1,..., x n X n 1 be any elements that are matching faces with respect to i. Then there exists an element w X n such that d j w = x j for j i. A fibrant -group means a -group which is a fibrant as a -set. A property of fibrant -group is as follows: Proposition 2.7. [2, Proposition 4.1.3] Let G = {G n } n 0 be a fibrant -group. Then (1) Bd n G is a normal subgroup of Z n G for each n; (2) d i ( j i Ker(d j : G n+1 G n )) = Bd n G for each 0 i n + 1; (3) π n (G) is an abelian group for n 1. Lemma 2.8. Let φ n : P n P n be the automorphism given by φ n (x) = σ n 1 xσ 1 Then, in P, φ n 1 d i if i < n 1 d i φ n = d n if i = n 1 d n 1 if i = n Proof. For 1 i < j n, φ n (A i,j ) = σ n 1 A i,j σn 1 1 = σ n 1 σ j 1 σ j 2 σ i+1 σi 2σ 1 i+1 σ 1 j 1 σ 1 n 1 A i,j if 1 i < j < n 1 = A i,n if 1 i < j = n 1 A n 1,n A i,n 1 A 1 n 1,n if 1 i < j = n. n 1.

15 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 15 Now φ n (A 0,j ) = σ n 1 A 0,j σn 1 1 = σ n 1 (σ j σ j+1 σ n 2 σn 1σ 2 n 2 σ j ) 1 (σ j 1 σ 2 σ1σ 2 1 σ j 1 ) 1 σn 1 1 A 0,j if j < n 1 = A 0,n if j = n 1 A n 1,n A 0,n 1 A 1 n 1,n if j = n. For k > 0, using the crossed simplicial structure on B, we obtain d k φ n (A i,j ) = d k (σ n 1 A i,j σn 1 1 ) = d k σ n 1 d k σn 1 A i,j d k (σn 1A i,j)(σn 1 1 ) = d k σ n 1 d k σn 1 A i,j d k σn 1 (σn 1 1 ) σ n 2 d k A i,j σn 2 1 = φ n 1(d k A i,j ) if k < n 1 = d n A i,j if k = n 1 d n 1 A i,j if k = n. For k = 0, we have φ n 1 d 0 (A i,j ) = φ n 1 (A i 1,j 1 ) = and hence the result. A i 1,j 1 if 1 i 1 < j 1 < n 2 A i 1,n 1 if 1 i 1 < j 1 = n 2 A n 2,n 1 A i,n 2 A 1 n 2,n 1 if 1 i 1 < j 1 = n 1. A 0,j if 0 = i 1 < j 1 < n 2 A 0,n 1 if 0 = i 1 < j 1 = n 2 A n 2,n 1 A 0,n 2 A 1 n 2,n 1 if 0 = i 1 < j 1 = n 1, A i 1,j 1 if 1 i < j 1 < n 1 A i 1,n 1 if 1 i < j = n 1 = A n 2,n 1 A i,n 2 A 1 n 2,n 1 if 1 i < j = n. = d 0 φ n (A i,j ) Proposition 2.9. The -group P is fibrant. Proof. Let x 0,..., x i 1, x i+1,..., x n be elements in P n 1 which match faces with respect to i. Since P 0 = P 1 = 1, we may assume that n 1 2. Case I. i < n. Note that the last degeneracy s n 1 : P n 1 P n is defined. Let y j = x j (d j s n 1 x n ) 1 for 0 j n and j i. Then the elements y 0,..., y i 1, y i+1,..., y n match faces. Note that y n = x n (d n s n 1 x n ) 1 = x n x 1 n = 1 and d n 1 y j = d n 1 x j (d n 1 d j s n 1 x n ) 1 = d n 1 x j (d j d n s n 1 x n ) 1 = d n 1 x j (d j x n ) 1 = 1 ( because x j matching faces ) for j n 1 with j i. Thus y 0,..., y i 1, y i+1,..., y n 1 match faces with respect to i in ( ΩP) n 2 = F [S 1 ] n 2. Since F [S 1 ] admits a simplicial group structure, it is

16 16 JINGYAN LI AND JIE WU fibrant. Thus there exists w ( ΩP) n 1 = Ker(d n : P n P n 1 ) such that d j w = y j for 0 j n 1 with j i. It follows that d n (ws n 1 x n ) = x n and d j (ws n 1 x n ) = d j (w)d j s n 1 (x n ) = y j d j s n 1 (x n ) = x j (d j s n 1 x n ) 1 d j s n 1 (x n ) = x j for 0 j n 1 with j i. Case II i = n. Let x 0,..., x n 1 P n 1 match faces with respect to n. We may assume that n 4 because P 0 = P 1 = 1 and P 2 = ( ΩP) 1 = F [S 1 ] 1 and F [S 1 ] is fibrant. We check that y 0 φ n 1 (x 0 ), y 1 = φ n 1 (x 1 ),..., y n 2 = φ n 1 (x n 2 ), y n = x n 1 match faces with respect to n 1. For 0 j < i n 2, Moreover, for 0 j n 2, d j y i = d j φ n 1 (x i ) = φ n 2 d j (x i ) = φ n 2 d i 1 x j = d i 1 φ n 1 x j = d i 1 y j. d j y n = d j x n 1 = d n 2 x j = d n 1 φ n 1 x j = d n 1 y j. Thus y 0, y 1,..., y n 2, y n match faces with respect to n 1. From Case I, there exists an element w P n such that d j w = y j for j n 1. Consider the element φ 1 n (w) P n. For 0 j n 2, φ n 1 d j φ 1 n w = d j φ n φ 1 n w = d j w = y j = φ n 1 x j and so d j φ 1 n w = x j. Since d n φ n = d n 1, we have d n = d n 1 φ 1 n and so The proof is finished now. x n 1 = y n = d n w = d n 1 φ 1 w. Proposition 2.10 (Central Extension Theorem). For each n, π n (P) is contained in the center of P n /Bd n P. Proof. By definition, π n (P) = Z n P/Bd n P. It suffices to show that the commutator [x, A i,j ] Bd n P for any x Z n P and 1 i < j n. Since d j x = 1 for all 0 j n. The elements x 0 = 1, x 1 = 1,..., x i 1 = 1, x i = x, x i+2 = 1,..., x n+1 = 1 are matching faces with respect to i + 1. Thus there exists an element y P n+1 such that d j y = x j for j i + 1, that is, d j y = 1 for j i, i + 1 and d i y = x. Let w = [y, A i+1,j+1 ]. Then d j w = [d j y, d j A i+1,j+1 ] = Thus [x, A i,j ] Bd n P and hence the result. [1, d j A i+1,j+1 ] = 1 if j i, i + 1 [x, A i,j ] if j = i [d i+1 y, d i+1 A i+1,j+1 ] = [d i+1 y, 1] = 1 if j = i + 1.

17 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 17 For a group G, denote by G ab the abelianization of G. By taking the abelianization of the Moore chain complex (NP, d 0 ), we have the following application of the Central Extension Theorem. Theorem There is a decomposition for each n, where A n is torsion free. H n ((NP) ab ) = π n (S 2 ) A n Proof. By Proposition 2.10, there is a central extension Z n P/Bd n P N n P/Bd n P π Bdn 1 P = d 0 (N n P). Since Bd n 1 P is a subgroup of the free group Ker(d n 1 : P n 1 P n 2 ), Bd n 1 P is a free group. By taking a cross-section Bd n 1 N n P/Bd n P, it follows that there is a product decomposition (2.14) N n P/Bd n P = π n (S 2 ) Bd n 1 P because Z n P/Bd n P lies in the center of N n P/Bd n P. Thus there is a decomposition (2.15) (N n P/Bd n P) ab = πn (S 2 ) (Bd n 1 P) ab. Now from the commutative diagram of chain complexes of groups N n+1 P d 0 N n P d 0 N n 1 P d 0 we have q (N n+1 P) ab d0 (N n P) ab and so the composite q q d0 (N n 1P) ab d0, q(bd n P) = Im(d 0 : (N n+1 P) ab (N n P) ab ) N n P q (Nn P) ab (Nn P) ab / Im(d 0 : (N n+1 P) ab (N n P) ab ) factors through the quotient group (N n P/Bd n ) ab of N n P. On the other hand, let p: N n P (N n P/Bd n ) ab be the quotient homomorphism. Then p factors through the abelianization q : N n P (N n P) ab. Since the composite N n+1 P d0 Nn P p (Nn P/Bd n ) ab is trivial, the quotient map p: N n P (N n P/Bd n ) ab factors through It follows that (N n P) ab / Im(d 0 : (N n+1 P) ab (N n P) ab ). (2.16) (N n P) ab / Im(d 0 : (N n+1 P) ab (N n P) ab ) = (N n P/Bd n ) ab. By definition, H n ((NP) ab ) = Ker(d 0 : (N n P) ab (N n 1 P) ab )/ Im(d 0 : (N n+1 P) ab (N n P) ab ). Let q : (N n P/Bd n ) ab π n (S 2 ) be the projection from Decomposition Then the composite π n (S 2 ) = H n (NP) q Hn ((NP) ab ) (N n P/Bd n ) ab q πn (S 2 )

18 18 JINGYAN LI AND JIE WU is an isomorphism. Thus there is a decomposition H n ((NP) ab ) = π n (S 2 ) A n. From Decomposition 2.15, A n is isomorphic to a subgroup of the free abelian group Bd n 1 P. The assertion follows. Remark By inspecting the proof, the only required assumptions for the -group P are: (1). Bd n P is a normal subgroup of N n P, (2). π n (P) = Z n P/Bd n P is contained in the center of N n P/Bd n P and (3). Bd n 1 P is a free group. Thus Theorem 2.11 can be generalized to any -groups satisfy the above conditions. By using the Central Extension Theorem in [34] and Kan s free group construction [20] for any loop spaces, one obtains a result that the homotopy groups of any simply connected space are the summands of the homology of an abelian chain complex. The further investigation for general spaces will be given in our subsequent work. Corollary For each n, there is a decomposition for some group B n. H n (Z(NP)) = π n (S 2 ) B n Proof. The assertion follows from the fact that the abelianization NP (NP) ab factors through the group ring Z(N P). Corollary There is an isomorphism of groups for n 4. Tor(H n ((NP) ab )) = π n (S 2 ) Proof. The assertion follows from Serre s result [30] that π n (S 2 ) is a finite group for n 4. Corollary Let n 4. Then (1). For a prime p > 2, the p-torsion component of π n (S 2 ) is given by the image of the Bockstein on the mod p homology H (Brun ab Z Z/p). (2). For p = 2, The 2-torsion component Tor 2 (π (S 2 )) is then determined by the images of the first and the second Bockstein on the mod 2 homology H (Brun ab Z Z/2). Proof. Assertion (1) follows from Selick s Theorem [28] that p Tor p (π n (S 2 )) = 0 for n 4, and assertion (2) follows from the fact that 4 Tor 2 (π n (S 2 )) = 0 by James Theorem [17]. Let δ n = (σ 1 σ n 1 ) n = (A 1,2 )(A 1,3 A 2,3 ) (A 1,n A 2,n A n 1,n ) be the generator for the center Z(P n ) of P n with n 2. Proposition The center of P n /Bd n P is contained in π n (P) Z for n 3.

19 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 19 Proof. First we prove the assertion for n 4. Let x P n such that, modulo Bd n P, x lies in the center P n /Bd n P. Then d n x Z(P n 1 ) and so there exists an integer q such that d n x = δ q n 1. Let y = δ q n x. Then d n y = 1. Observe that ( ΩP) n 1 = Ker( d n : P n P n 1 ) with Bd n P ( ΩP) n 1. Since y = δn q x lies in the center of P n /Bd n P by modulo Bd n P, y lies in the center of ( ΩP) n 1 /Bd n P. Note that ΩP = F [S 1 ] as simplicial group. By [34, Theorem 2.18], y π n 1 ( ΩP) = π n (P) by modulo Bd n P and hence the result. 3. Brunnian Braids and Boundary Brunnian Braids 3.1. The conjugation action of braids on P. Let χ σk : P n P n, β σ k βσ 1 k, be the conjugation action of σ i on P n. Part of the following lemma was given in [25]. Lemma 3.1. For 1 k n 1 and 0 i < j n, σ k A i,j σ 1 k = A i,j if k i 1, i, j 1, j A i,k+1 if j = k A 1 i,k+1 A i,ka i,k+1 if j = k + 1 and i < k A k,k+1 if j = k + 1 and i = k A i+1,j if i = k < j 1 A 1 k+1,j A k,ja k+1,j if i = k + 1. Proof. The proof follows by drawing the pictures of braids for examining the formulae. The case for j = k + 1 and 0 < i < k is pictured as follows, where one can turn (k + 1)st strand round on the right braid to catch the left braid. i k k+1 i k k+1 = σ k A i,k+1 σ k -1 A i,k+1-1 A i,k A i,k+1

20 Similarly one can draw the pictures for showing the case when i = k + 1. The case for j = k +1 and i = 0 is pictured below, where one can turn (k +1)st strand round 20 carefully on the right braid tojingyan catch the LI left ANDbraid. JIE WU k k+1 k k+1 = σ k A 0,k+1 σ k -1 A 0,k+1-1 A 0.k A 0,k+1 The examination of the rest cases is straightforward. Let θ : P n P n be the automorphism defined in Lemma 2.3. Proposition 3.2. The following property holds in P: (1). d 0 χ σk = χ σk 1 d 0 for each k 2. (2). d 0 χ σ1 = θd 0. (3). Let B n act on {1, 2,..., n} via its quotient to the symmetric group S n. Then d P i χ σ k = χ d B i 1 σ k d i σk for k, i 1. Proof. Note that d 0 A i,j = A i 1,j 1 for j > i 1. Assertion (1) follows from Lemma 3.1 by checking the formula that d 0 χ σk (A i,j ) = χ k 1 d 0 (A i,j ) for j > i 1 and k 2. (2). By Lemmas 2.3 and 3.1, d 0 χ σ1 (A i,j ) = θd 0 (A i,j ) for j > i 2. Now, by Lemma 3.1, { A1,j 1 if j > 2 d 0 χ σ1 (A 1,j ) = A 0,1 if j = 2. On the other hand, for j > 2, θd 0 (A 1,j ) = θ(a 0,j 1 ) = θ ( (A j 1,j A j 1,j+1 A j 1,n ) 1 (A 1,j 1 A 2,j 1 A j 2,j 1 ) 1) = θ ( (A j 1,j A j 1,n ) 1 (A 2,j 1 A j 2,j 1 ) 1) θ(a 1,j 1 ) 1 = (A j 1,j A j 1,n ) 1 (A 2,j 1 A j 2,j 1 ) 1 (A 1,j 1 ) 1 A 1 0,j 1 A 1,j 1 = A 1,j 1. Note that, in P n, (3.1) Thus A 0,1 A 0,2 A 0,3 A 0,n = d 0 (A 1,2 A 1,3 A 1,n+1 ) = d 0 (A 1 0,1 ) = d 0 d 0 (A 1 = d 0 d 1 (A 1 = 1. 1,2 ) 1,2 ) θd 0 (A 1,2 ) = θ(a 0,1 ) = θ(a 0,2 A 0,3 A 0,n ) 1 = (A 1,2 A 1,3 A 1,n ) 1 = A 0,1 = d 0 χ σ1 (A 1,2 )

21 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 21 and so assertion (2). (3). We can use the crossed simplicial group structure of B for computing the faces d P i on conjugations for i > 0. Let β be an element in P n. Then d P i (σ kβσ 1 k ) = db i 1 (σ kβσ 1 k ) = d B i 1 (σ k)d B (i 1) σ k (βσ 1 k ) = d B i 1 (σ k)d B (i 1) σ k (β)d B (i 1) (σ k β) (σ 1 = d B i 1 (σ k)d B (i 1) σ k (β)d B (i 1) σ k (σ 1 k ), where B n acts on {0, 1,..., n 1} via the symmetric group S n. From 1 = d B i 1 (σ kσ 1 k ) = d B i 1 (σ k)d B (i 1) σ k (σ 1 k ), we have d B (i 1) σ k (σ 1 k ) = (db i 1 (σ k)) 1. Thus d P i χ σ k (β) = d P i (σ kβσ 1 k ) = d B i 1 (σ k)d B (i 1) σ k (β)(d B i 1 (σ k)) 1 = d B i 1 (σ k)d P i σ k (β)(d B i 1 (σ k)) 1 = χ d B i 1 (σ k )d P i σ k (β) k ) and hence the result. Proposition 3.3. The action of θ on P n satisfies the following braiding identities: (1). θχ σ1 θ = χ σ1 θχ σ1. (2). θχ σi = χ σi θ for i > 1. Proof. By the proof of Proposition 3.2, θ(a 0,j ) = A 1,j for j 2. (1) We check the identity χ σ1 θχ σ1 (A i,j ) = θχ σ1 θ(a i,j ) holds for each 1 i < j n. It suffices to check that cases i = 1, 2 because A i,j are fixed under the action of χ σ1 and θ for i > 2. When i = 2, we have χ σ1 (A 2,j ) = A 1 2,j A 1,jA 2,j and so χ σ1 θχ σ1 (A 2,j ) = χ σ1 θ(a 1 2,j A 1,jA 2,j ) On the other hand, = χ σ1 (A 1 2,j A 1 1,j A 0,jA 1,j A 2,j ) = A 1 2,j A 1 1,j A 0,jA 1,j A 2,j ( because A 0,j and A 1,j A 2,j are fixed under χ σ1 ). θχ σ1 θ(a 2,j ) = θχ σ1 (A 2,j ) = θ(a 1 2,j A 1,jA 2,j ) = A 1 2,j A 1 1,j A 0,jA 1,j A 2,j. Thus χ σ1 θχ σ1 (A 2,j ) = θχ σ1 θ(a 2,j ). Similar computations apply to the case when i = 1, where θ(a 0,j ) = A 1,j for j 2 is used in this case. Assertion (1) follows. (2). For i 2, since the subgroup A s,t 2 s < t n is invariant under the action of χ σi and θ(a s,t ) = A s,t for 2 s < t n, we have χ σi θ(a s,t ) = θχ σi (A s,t ) for 2 s < t n. Now consider the case s = 1. For t i, i + 1 with t > 1, χ σi θ(a 1,t ) = χ σi (A 1 1,t A 0,tA 1,t ) = A 1 1,t A 0,tA 1,t = θχ σi (A 1,t ).

22 22 JINGYAN LI AND JIE WU For t = i, we have θχ σi (A 1,i ) = θ(a 1,i+1 ) = A 1 1,i+1 A 0,i+1A 1,i+1 and χ σi θ(a 1,i ) = χ σi (A 1 1,i A 0,iA 1,i ) = A 1 1,i+1 A 0,i+1A 1,i+1. Thus θχ σi (A 1,t ) = χ σi θ(a 1,t ) for t i + 1. For the case i = i + 1, we check another element A 0,i+1 with the equations θχ σi (A 0,i+1 ) = θ(a 1 0,i+1 A 0,iA 0,i+1 ) = A 1 1,i+1 A 1,iA 1,i+1 and χ σi θ(a 0,i+1 ) = χ σi (A 1,i+1 ) = A 1 1,i+1 A 1,iA 1,i+1. Thus θχ σi (A 0,i+1 ) = χ σi θ(a 0,i+1 ). Since A 0,i+1 = (A i+1,i+2 A i+1,i+3 A i+1,n ) 1 (A 1,i+1 A 2,i+1 A i 1,i+1 ) 1, A 1,i+1 = A 1 0,i+1 (A i+1,i+2a i+1,i+3 A i+1,n ) 1 (A 2,i+1 A 3,i+1 A i 1,i+1 ) 1 and so χ σi θ(a 1,i+1 ) = θχ σi (A 1,i+1 ). The assertion follows. From the above two propositions, we have the following. Corollary 3.4. There is a representation Φ: B n+1 Aut(P n ) given by Φ(σ 1 ) = θ, Φ(σ i ) = χ σi 1 for i > 1. Moreover there is commutative diagram P n+1 χ β Pn+1 for each β B n+1. P n d P 0 dp 0 Φ(β) Pn 3.2. Brunnian Braids. Recall that the Brunnian subgroup Brun n of B n is defined by n Brun n = Ker(d P i : P n P n 1 ). i=1 Consider the Fadell-Neuwirth fibration F (C Q n 1 ) F (C, n) p 1 F (C, n 1), where p 1 (x 1,..., x n ) = (x 2,..., x n ) and Q n 1 is the basepoint of F (C, n 1) consisting of a fixed choice of (n 1) distinct points in C. Since d 1 : P n P n 1 is induced by p 1 : P n = π 1 (F (C, n)) P n 1 = π 1 (F (C, n 1)) under a fixed choice of basepoint (q 1,..., q n ) of F (C, n) with q i R ordered by q 1 < < q n, Ker(d 1 ) is the free group of rank n 1 generated by A 1,j for 2 j n. Let B n 1 be the subgroup of B n generated by σ i for 2 i n. By Lemma 3.1, the conjugation action of B n 1 on Ker(d 1 ) is given by χ σk (A 1,j ) = Thus we have the following: A 1,j if j k, k + 1 A 1,k+1 if j = k A 1 1,k+1 A 1,kA 1,k+1 if j = k + 1.

23 ARTIN BRAID GROUPS AND HOMOTOPY GROUPS 23 Proposition 3.5. The conjugation action of Bn 1 on Ker(d 1 ) with the free generators {A 1,j } 2 j n is the Artin representation of B n 1 = Bn 1. Let F n 1 denote the subgroup Ker(d 1 ) of B n with the free generators A 1,j for 2 j n. Then Brun n F n 1 is a subgroup of Fn 1. By varying n, Fn 2 is regarded as the subgroup of Fn 1 generated by A 1,j for 2 j n 1. Then Brun n 1 is regarded as the subgroup of F n 1 in the following canonical way: Brun n 1 F n 2 F n 1. Consider the faces d i = d P i : P n P n 1 for 2 i n. Then A 1,j if j < i d i A 1,j = 1 if j = i A 1,j 1 if j > i. The group homomorphism d i Fn 1 : Fn 1 F n 2 can be described by the following table A 1,2 A 1,3 A 1,i 1 A 1,i A 1,i+1 A 1,n (3.2) d i = A 1,2 A 1,3 A 1,i 1 1 A 1,i A 1,n 1 for 2 i n. Let G be a group. The commutator [a, b] is defined by [a, b] = a 1 b 1 ab for a, b G. Lemma 3.6. Let n 3 and let S be a set of generators for Brun n 1. Then Brun n is the normal subgroup of F n 1 generated by the commutator for x S. [x, A 1,n ] Proof. The assertion holds for n = 3 because Brun 3 is the commutator subgroup of F 2, which is normally generated by [A 1,2, A 1,3 ]. Thus we may assume that n > 3. Let C be the normal subgroup of F n 1 generated by [x, A 1,n ] for x S. For each x S Brun n 1, we have d i x = 1 for 2 i n 1. Thus d i [x, A 1,n ] = [d i x, d i A 1,n ] = [1, d i A 1,n ] = 1 for x S and 2 i n 1. Since d n A 1,n = 1, we also have d n [x, A 1,n ] = 1 for x S. Since Brun n is a normal subgroup of F n 1, C Brun n. For proving that C = Brun n, it suffices to show that the composite Brun n Fn 1 Fn 1 /C is trivial. Let w Brun n. Since F n 1 is the free product of F n 2 and the subgroup Z(A 1,n ) of F n 1 generated by A 1,n, w admits a unique decomposition such that (1). w j F n 1 for 1 j t, (2). l j Z for 1 j t, (3). w j 1 for 1 j t 1, (4). l j 0 for 2 j t. w = A l1 1,n w 1A l2 1,n w 2 A lt 1,n w t

24 24 JINGYAN LI AND JIE WU For each 2 i n 1, we have 1 = d i w = A l1 1,n 1 d i(w 1 )A l2 1,n 1 d i(w 2 ) A lt 1,n 1 d i(w t ) in F n 2. From Equation 3.2, d i (A 1,j ) F n 3 for j n 1. Thus d i ( F n 2 ) F n 3. It follows that d i (w j ) F n 3 for 1 j t. Since F n 2 = F n 3 Z(A1,n 1 ) is the free product, we have the equations d i (w j ) = 1 for 2 i n 1 and 1 j t, that is, (3.3) w j Brun n 1 for each 1 j t, and A l1+l2+ +lt 1,n 1 = 1, that is, (3.4) l 1 + l l t = 0. Next we apply the last face d n to w, from Equation 3.2, we have d n (w j ) = w j for each j and d n (A 1,n ) = 1. Hence we have the equation (3.5) w 1 w 2 w t = 1. For a, b F n 1, we write a b if the images of a and b in the quotient group F n 1 /C are the same. By the definition of C, [x, A 1,n ] 1 for x S and so xa 1,n A 1,n x for any x S. Since S generates Brun n 1, we have (3.6) va 1,n A 1,n v for any word v Brun n 1. It follows that and so w C. This finishes the proof. w = A l1 1,n w 1A l2 1,n w 2 A lt 1,n w t A l1+l2+ +lt 1,n w 1 w 2 w t = 1 Now define a subset T n of P n by setting T 2 = {A 1,2 } and T n = {[[[A 1,2, A i2,3], A i3,4],..., A in 1,n] 1 i t t for t = 2, 3,..., n 1}. Note that d P k [[[A 1,2, A i2,3], A i3,4],..., A in 1,n] = 1 for each k 1. T n is a subset of Brun n. Theorem 3.7. For each n 2, Brun n is the normal subgroup of P n generated by T n. Proof. Let D n be the normal subgroup of P n generated by T n. Since T n Brun n and Brun n is a normal subgroup of P n, D n Brun n. We show that D n = Brun n by induction on n. Clearly the assertion holds for n = 2, 3. Suppose that the assertion holds for n 1, that is, D n 1 = Brun n 1 with n > 3. Let x y denote the conjugation yxy 1 for x, y in a group G. For x, y P n, write x y if the images of x and y in the quotient group P n /D n are the same. Let S = {a β β P n 1, a T n 1 }.