A BRAIDED SIMPLICIAL GROUP. 1. Introduction. In this article, we study the homotopy groups by considering the braid group

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1 A BRAIDED SIMPLICIAL GROUP JIE WU Abstract. By studying the braid group action on Milnor s construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group. 1. Introduction In this article, we study the homotopy groups by considering the braid group actions on simplicial groups. The purpose is to establish a relation between the braid group actions on certain combinatorially given groups and the homotopy groups of the 3-sphere. We first recall a combinatorial description of the homotopy groups of the 3-sphere in [16]. Let F (x 1,..., x n ) be the free group generated by the letters x 1,..., x n. Let w(x 1,..., x n ) = x ɛ 1 i1 x ɛt i t be a word. Given a 1,..., a n F (x 1,..., x n ), we write w(a 1,..., a n ) F (x 1,..., x n ) for a ɛ 1 i1 a ɛ t i t. The group G(n) is defined to be the quotient group of F (x 1,..., x n ) modulo the following relations: (R 1 ) the product x 1 x n ; Research is supported in part by the Academic Research Fund of the National University of Singapore RP

2 2 JIE WU (R 2 ) the words w(x 1,..., x n ) that satisfy: w(x 1,..., x i 1, 1, x i+1,..., x n ) = 1 for 1 i n. Relations R 2 consist of all of words that will be trivial if one of the generators is replaced by the identity element 1. The smallest normal subgroup of F (x 1,..., x n ) that contains relations R 1 and R 2 was determined as a subgroup of F (x 1,..., x n ) generated by certain systematic and uniform iterated commutators [16]. Theorem 1.1. [16] For n 3, the homotopy group π n (S 3 ) is isomorphic to the centre of G(n). A natural question that arises from Theorem 1.1 is how to give a group-theoretical approach to the homotopy groups, that is, how to understand the centre of the group G(n). Let B n be the braid group on n strings. There is a canonical braid group action on G(n) that is induced by the Artin representation of the braid group B n on the free group F (x 1,..., x n ). (We will go over the definition of B n and Artin s representation in Section 2.) These actions give a canonical homomorphism from the braid group B n into the automorphism group of G(n). Since Quillen s plus construction of the classifying space for the stable braid group is (up to homotopy type) the double loop space of the 3-sphere [5], these braid group actions do not appear to be a coincidence. Larry Taylor conjectured that the centre of G(n) is the

3 A BRAIDED SIMPLICIAL GROUP 3 fixed set of the braid group action on G(n). We solve Taylor s conjecture as follows. Let K n be the pure braid group, that is, K n is the kernel of the canonical epimorphism from B n to the symmetric group Σ n. In geometry, the group K n is the fundamental group of the configuration space F (R 2, n), where F (M, n) = {(x 1,..., x n ) M n x i = x j for i = j} for any manifold M (See [5]). Let Z(G(n)) be the centre of G(n). Theorem 1.2. Let n 4. Then 1) the fixed set of the pure braid group action on G(n) is the centre of G(n) and so is π n (S 3 ); 2) the fixed set of the braid group action on G(n) is the subgroup {x Z(G(n)) 2x = 0}. We should point out that the determination of the fixed set of K n -action on G(n) by (combinatorial) group-theoretic means seems beyond the reach of current techniques. On the other hand, braid group actions have been much studied in several areas such as group theory and low-dimensional topology. Various problems arising from physics are related to braid group actions as well. Theorem 1.2 suggests that the homotopy groups play a certain role for braid group actions. In the range in which π (S 3 ) is

4 4 JIE WU known ( 55, see [9, 15]), by homotopy-theoretic means, we might gain insight into these difficult group-theoretic questions. The B n -action on G(n) induces a homomorphism θ : B n Aut(G(n)), where Aut(G) is the automorphism group of G. Fred Cohen asked what is the kernel of the map θ. We answer Cohen s question by determining the kernel of θ. We find that the kernel of θ is related to so-called Brunnian braids in low-dimensional topology. Recall that a (geometric) braid β on n-strings is called Brunnian if β becomes to be a trivial braid when any one of the strings is removed. A link obtained by closing up a Brunnian braid is called a Brunnian link. An example of a 3-component Brunnian link is the well-known Borromean Ring. Let C n be the set of Brunnian braids on n strings. Then C n is a normal subgroup of B n. Recall that the centre of B n is isomorphic to the infinite cyclic group Z for n 3. (See [4, Corollary 1.8.4]). Theorem 1.3. Let θ n : B n Aut(G(n)) be the representation map induced by the B n -action on G(n). Suppose that n 4. Then the kernel of θ n is the subgroup of B n generated by Z(B n ) and C n and so Ker(θ n ) = Z(B n ) C n = Z Cn. The group C n has been largely studied in low dimensional topology with various important results. There is a 30-year-old problem in low-dimensional topology proposed by J. Birman: how to find a free basis for R n C n [4, Problem 23, pp.219], where R n

5 A BRAIDED SIMPLICIAL GROUP 5 is the kernel of K n = π 1 (F (R 2, n)) π 1 (F (S 2, n)) induced by the canonical inclusion F (R 2, n) F (S 2, n). Birman s problem still remains open in general. Recently A. J. Berrick, Y. L. Wong and the author showed that the subgroup R n C n of B n is isomorphic to the cycles in dimension n 1 of the Milnor construction F (S 1 ) [3]. This gives a relation between these special Brunnian braids and the homotopy groups. We should point out that the group R n C n plays an important role for the Burau and Gassner representations of braids. (See [4].) It was known that the sequence of pure braid groups, K = {K n+1 } n 0, forms a simplicial group where the i-th face is given by deleting the i-string and the i- degeneracy is given by doubling the i-string. (See for instance [10].) Recently Fred Cohen and the author showed that F (S 1 ) embeds into K as a simplicial subgroup [6]. From this we obtained certain physical interpretations of the Lambda algebra and the group G(n). The article is organized as follows. In section 2, we study the braid group action on the Milnor construction of the simplicial 1-sphere. A relation between the simplicial structure and the braid group action is given in Proposition 2.1. After we establish the systematic relations, braided simplicial groups are introduced in this section. Then we discuss some basic properties of braided simplicial groups such as the simplicial loop group and the Moore-Postnikov system. Theorem 2.10 and Proposition 2.11 give a relation between the fixed set of the braid group action and the homotopy

6 6 JIE WU groups for a general braided simplicial group. Theorem 1.2 follows from Lemma 2.8 and Theorem In section 3, we give a geometrical description of the group G(n). The proof of Theorem 1.3 is given in this section. The author would like to thank Professors Fred Cohen and Jon Berrick for their helpful suggestions and kind encouragement. The author also would like to thank the referee for his/her many important and useful suggestions and comments. The manuscript has been substantially modified along the referee s suggestions. In particular, the content in Section 3 of the original manuscript has been replaced by the new material given here. 2. Braid Group Actions on F (S 1 ) 2.1. The braid groups. In this subsection, we go over some terminology of the braid groups. A reference for braid groups is Birman s book [4]. There are several equivalent definitions of the braid group B n. A combinatorial definition is as follows: (due to Artin [1]) The group B n is generated by letters σ 1, σ 2,..., σ n 1 with relations σ i σ j = σ j σ i for i j 2 and σ i+1 σ i σ i+1 = σ i σ i+1 σ i for 1 i n 1. For geometric reasons, B n is called the braid group on n strings. Recall that the symmetric group Σ n is the quotient group of B n by the additional relation that σi 2 = 1 for each i. Let F n = F (x 1,..., x n ) be the free group generated by letters x 1,..., x n.

7 A BRAIDED SIMPLICIAL GROUP 7 The Artin representation of B n is a homomorphism φ from B n to Aut(F n ), where x i+1 if j = i, (1) φ(σ i )(x j ) = x 1 i+1 x ix i+1 if j = i + 1, x j otherwise, for 1 i n 1. By Artin s theorem [2], the map φ is a monomorphism. Usually we still write σ i for φ(σ i ) as an element in Aut(F n ). This gives another definition of B n, that is, B n is the subgroup of Aut(F n ) generated by σ i for 1 i n 1. Clearly the product element x 1 x 2 x n is a fixed point of the B n -action on F n. Thus the Artin representation induces a B n -action on ˆF n, where ˆF n is the quotient group of F n modulo the single relation x 1 x 2 x n = 1. As a group, ˆFn is a free group of rank n Braided simplicial groups. We refer to [8, 12] for terminology of simplicial sets and simplicial groups. Let be the category of finite ordered sets and ordered functions, where a function f is ordered if f(x) f(y) when x y. Let C be any category. The category has objects [n] = {0,, n} for n 0 and the morphisms in C are generated by the face functions d i : [n 1] [n] (misses i) and the degeneracy functions s i : [n] [n + 1] (hits i twice) for 0 i n. Recall that a simplicial object X over C is a contravariant functor from 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 )

8 8 JIE WU 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: 1) d j d i = d i d j+1 for j i; 2) s j s i = s i+1 s j for j i; s i 1 d j for j < i 3) d j s i = id for j = i, i + 1 s i d j 1 for j > i follow from the well-known formulas for functions d i and s j in the category. A simplicial object over sets (resp. monoids, groups, Lie algebras, spaces and etc) is called a simplicial set (resp. monoid, group, Lie algebra, space and etc.) Let X be a pointed set. We write J(X) ( resp. F (X)) for the free monoid (resp. free group) generated by X with the single relation that the base-point = 1. The functor J (resp. F ) is the coadjoint functor of the forgetful functor from monoids (resp. groups) to pointed sets. Let X be a pointed simplicial set. The simplicial monoid J(X) = J X is called the James construction of X and the simplicial group F (X) = F X is called the Milnor construction of X. The geometric realization F (X) of F (X) is a free group generated by X with the single relation that the base-point = 1 and with compactly generated (weak) topology. As a space, there is a homotopy equivalence F (X) ΩΣ X. The geometric realization of J(X) is the

9 A BRAIDED SIMPLICIAL GROUP 9 classical James construction on the pointed space X. We keep in mind that F (S n ) is a simplicial group model of ΩS n+1. Let F (S 1 ) be Milnor s construction of the simplicial 1-sphere S 1. Then F (S 1 ) n+1 = F (y 0,..., y n ) is a free group generated by letters y 0,..., y n with the following simplicial structure: y k 1 for j k, d j y k = 1 for j = k + 1, and y k for j > k + 1, y k+1 for j k, s j y k = y k y k+1 for j = k + 1, y k for j > k + 1, for 0 j n + 1, where y 1 = (y 0 y n 1 ) 1 in F (S 1 ) n (see [16, Lemma 4.1]). Let F n+2 = F (y 1, y 0,..., y n ) be the free group generated by the letters y 1, y 0,..., y n. Let B n+2 act on F n+2 in Artin s sense. We relabel the generators for B n+2 by σ 1, σ 0,..., σ n 1. Observe that the composite F (y 0, y 1,..., y n ) F n+2 = F (y 1, y 0,..., y n ) ˆFn+2 is an isomorphism. Thus the B n+2 -action on F n+2 induces a B n+2 -action on F (y 0,..., y n ), where the map σ i : F (y 0,..., y n ) F (y 0,..., y n ) is given by the equation (1)

10 10 JIE WU for i 0 and σ 1 has to be reformulated by σ 1 (y 0 ) = y0 1 y 1 y 0 = y0 1 yn 1 y1 1, σ 1 (y j ) = y j for j > 0. By direct calculation, we have Proposition 2.1. The following identities hold for the braid groups action on F (S 1 ): σ k 1 d j j k, (2) d j σ k = d k+2 j = k + 1, d k+1 j = k + 2, σ k d j j > k + 2; (3) s j σ k = σ k+1 s j j k, σ k+1 σ k s k+2 j = k + 1, σ k σ k+1 s k+1 j = k + 2, σ k s j j > k + 2. Motivated by this proposition, we give the following definition. Definition 2.2. A graded group means a sequence of groups G = {G n } n 0. A graded group G is called braided if each G n admits a B n+1 -representation, that is, a group homomorphism B n+1 Aut(G n ). Let G and H be braided (graded) groups. A braided homomorphism f : G H means a sequence of homomorphisms f n : G n H n such

11 A BRAIDED SIMPLICIAL GROUP 11 that each f n is B n+1 -equivariant. A braided simplicial group is a braided group that in addition satisfies the relations spelled out in Proposition 2.1. Note. It was known [10] that the sequence of the braid groups B = {B n+1 } n 0 is a crossed simplicial group. Recently F. Cohen and the author observed that a braided simplicial group G means a simplicial group G that admits a crossed B- representation [6], that is, the action B G G is not a simplicial map but satisfies certain crossed conditions. Roughly speaking, this action twists faces and degeneracies. Let G be a simplicial group and let NG be the Moore chain complex of G, that is, NG n = {x G n d j x = 1 for j > 0} with differential d 0 : NG n NG n 1, where d 2 0 = 1 follows from the simplicial identity d i d j = d j d i+1 for i j. Let Z(G) and B(G) be the sets of cycles and boundaries of G, respectively, that is, Z n (G) = {x G n d j x = 1 for j 0}, B n (G) = {d 0 x x N n+1 (G)}. By Moore s classical theorem [14], π n (G) = Z n (G)/B n (G), where π n (G) means the n-th homotopy group of the geometric realization G of G. Let G be a braided

12 12 JIE WU simplicial group. A subgroup H n of G n is called a braided subgroup if β(h n ) H n for any β B n+1, that is, H n is invariant under the B n+1 -action. Proposition 2.3. Let G be a braided simplicial group. Then the subgroups Z(G) and B(G) of G are braided. Proof. By Proposition 2.1(2), Z(G) is a braided subgroup. Now let x = d 0 y B(G) n, where y NG n+1. By Proposition 2.1(2), we have σ k x = σ k d 0 y = d 0 σ k+1 y for each k 1 and d j σ k+1 y = 1 for each j > 0 and k 1. Thus σ k x B(G) n for each k 1 and so B(G) is a braided subgroup, which is the assertion. Note: NG n is invariant under the subgroup of B n+1 generated by σ k with k 0. But it is not invariant under σ 1. Since there are relations d 0 σ 1 = d 1, d 1 σ 1 = d 0, and d j σ 1 = σ 1 d j for j > 1, we have

13 A BRAIDED SIMPLICIAL GROUP 13 Proposition 2.4. Let G be a braided simplicial group. Then Z(G) n = NG n σ 1 (NG n ). for each n. Now we study the braid group actions on the Postnikov system of a braided simplicial group G. Let I = (i 1,..., i k ) be a sequence of non-negative integers and let d I denote the composite of face homomorphisms d I = d i1 d ik. Given a simplicial group G, the simplicial sub groups R n G and R n (G) are defined as follows: R n G q = {x G q d I (x) = 1 for any I = (i 1,..., i q n )}, R n G q = {x G q d I (x) B(G) n for any I = (i 1,..., i q n )}. Let P n G = G/R n G and P n G = G/ R n G. Then {P n G} is the Postnikov system of G (see [8, 14]). The quotient homomorphism P n G P n G is a homotopy equivalence (see [16]). The tower P n G P n G P n 1 G

14 14 JIE WU is called a modified Postnikov system of G. An important property of the modified Postnikov system is that the short exact sequence of simplicial groups 0 K(π n (G), n) P n G P n 1 G 1 is a central extension for n 1 [16, Theorem 2.12]. By Propositions 2.1 and 2.3, we have Theorem 2.5. Let G be a braided simplicial group. Then, for each n, the simplicial quotient groups P n G and P n G are braided. Thus the modified Postnikov system of G is braided. In particular, there is a braided central extension 0 K(π n (G), n) P n (G) P n 1 (G) 1 for n 1. Let G be a braided simplicial group. Then R 0 G is a braided simplicial subgroup of G by the theorem above. Recall that the simplicial loop group ΩG of G is defined by ΩG n = Ker(d 0 ) R 0 G n+1 with d j (ΩG) = d j+1 (G) and s j (ΩG) = s j+1 (G) (see [8]). By Proposition 2.1, Ker(d 0 ) R 0 G n+1 is invariant under the action of σ j for 0 j n 1. Let B n+1 be the subgroup of B n+2 generated by σ 0,..., σ n 1. Then B n+1 = B n+1 under the canonical isomorphism that sends σ j to σ j 1 for 0 j n 1. Thus we obtained the following theorem.

15 A BRAIDED SIMPLICIAL GROUP 15 Theorem 2.6. Let G be a braided simplicial group. Then the simplicial loop group ΩG is braided. Thus any iterated simplicial loop groups of G are braided. Corollary 2.7. The loops and the modified Postnikov system of F (S 1 ) are braided Fixed Sets of Braided Actions. Let G be a simplicial group and let x, y G n be two elements. We say that x is homotopic to y, which is denoted by x y, if xy 1 B(G) n. (Note. If x y, then d j x = d j y for each j.) Lemma 2.8. Let G be a braided simplicial group and let x Z(G) n be a cycle with n 1. Then σ k (x) x 1 for each k 1. Proof. For each k 1, consider the element σ k+1 s k+1 x. By the identities (2) and (3) of Proposition 2.1, we have σ k d j s k+1 x = 1 for j < k + 1 σ k x for j = k + 1 d j σ k+1 s k+1 x = d k+3 s k+1 x = 1 for j = k + 2 d k+2 s k+1 x = x for j = k + 3 σ k+1 d j s k+1 x = 1 for j > k + 3.

16 16 JIE WU By the Homotopy Addition Theorem [8, Theorem 2.4], one gets σ k x x 1 and hence the result. Let G be a braided simplicial group and let S be a subset of G n. Define hstab Bn+1 (S) = {σ B n+1 σx x for all x S}. Since B(G) is invariant under the braid group action, hstab Bn+1 (S) is a subgroup of B n+1. Let B n+1 be the kernel of the composite B n+1 Σ n+1 sign Z/2, that is, Bn+1 is the pre-image of the alternating group A n+1. Lemma 2.9. Any normal subgroup of B n+1 which contains all elements of the form σ i σ j is either B n+1 or B n+1. Proof. Let P be a normal subgroup of B n+1 which contains all elements of the form σ i σ j. According to [11, Theorem N7, pp.173], the kernel K n+1 of the mapping B n+1 Σ n+1 is the normal subgroup generated by σ 2 1. Thus K n+1 P. Since the image of P in Σ n+1 contains all even permutations, we have Im(P Σ n+1 ) A n+1 and hence the result. Theorem Let G be a braided simplicial group. Then 1) hstab Bn+1 (Z(G) n ) = B n+1 or B n+1 ;

17 A BRAIDED SIMPLICIAL GROUP 17 2) hstab Bn+1 (Z(G) n ) = B n+1 if and only if 2 π n (G) = 0. Proof. Let σ hstab Bn+1 (Z(G) n ) and let x Z(G) n. By Lemma 2.8, we have (σ 1 k σσ k)(x) (σ k σ)(x 1 ) σ k (x 1 ) x for each k 1. Thus hstab Bn+1 (Z(G) n ) is a normal subgroup of B n+1. By Lemma 2.8, we have σ s σ t hstab Bn+1 (Z(G) n ) for any s, t 1. It follows that hstab Bn+1 (Z(G) n ) = B n+1 or B n+1, which is the assertion 1. If hstab Bn+1 (Z(G) n ) = B n+1, then by Lemma 2.8 x x 1 for any x Z(G) n. Thus 2 π n (G) = 0. Conversely, if 2 π n (G) = 0, then σ k (x) x for any x Z(G) n and k 1. Thus hstab Bn+1 (Z(G) n ) = B n+1. This shows assertion 2. Let G be a braided simplicial group and let H be a subgroup of B n+1. Define (G n ) h H = {x G n σ(x) x for all σ H},

18 18 JIE WU that is, (G n ) h H is the homotopy fixed set of H. Then (G n ) h H is a subgroup of G n. By Theorem 2.10, we have that (G n ) h B n+1 Z(G) n. Proposition Let G be a braided simplicial group and let x G h B n+1 n with n 2. Then 1) d j (x) = d j+2 (x) for each j; 2) σ k d j (x) = d j+1 (x) for each j, k; 3) d j (x) is a fixed point of B n for each j. In particular, if d j x = 1 for some j, then x Z(G) n. Proof. Since x (G n ) h B n+1, we have σ s σ t (x) x for any s, t 1. It follows that σ 1 x σ 0 x σ 1 x... σ n 2 x. Now for each 1 k n 3, we have d k+1 x = d k+2 σ k x = d k+2 σ k+1 x = d k+3 x and so assertion 1 follows.

19 Now for each 0 s n 2, we have A BRAIDED SIMPLICIAL GROUP 19 d s+1 (x) = d s σ s 1 (x) = d s σ s (x) = σ s 1 d s (x). Let j and k be integers with 0 j n 1 and 1 k n 3. If j k + 1 mod 2, then σ k d j (x) = σ k d k+1 x = d k+2 x = d j+1 x by assertion 1. If j k mod 2 with k 0, then σ k d j (x) = σ k d k (x) = d k σ k+1 (x) = d k σ k 1 (x) = d k+1 (x) = d j+1 (x) by assertion 1. Assume that j k mod 2 with k = 1. Then σ 1 d j x = σ 1 d 1 x = σ 1 d 0 σ 1 x = d 0 (σ 0 σ 1 (x)) = d 0 x = d j+1 x. Assertion 2 follows. For any s, t 1, we have σ s (σ t d 0 (x)) = σ s (d 0 (σ t+1 x)) = d 0 (σ s+1 σ t+1 x) = d 0 x. Thus σ t σ s (d 0 (x)) = d 0 (x) for any s, t 1 and so d 0 (x) is a fixed point of B n. By assertions 1 and 2, we have σ t (σ s (d 1 (x))) = σ t (d 2 (x)) = σ t (d 0 (x)) = d 1 (x)

20 20 JIE WU and so d 1 (x) is a fixed point of B n. Assertion 3 follows. Let B n be the subgroup of B n+1 generated by σ j with j 0. Let B n be the normal subgroup of B n generated by σ i σ j for i, j 0. By inspecting the proof, we have Proposition Let G be a braided simplicial group and let x (G n ) h B n with n 2. Then 1) d j (x) = d j+2 (x) for each j 1; 2) σ k d j (x) = d j+1 (x) for each j, k 1; 3) d j (x) is a fixed point of B n 1 for each j; 4) d 0 (x) is a fixed point of B n. In particular, if d 0 x = 1 and d j x = 1 for some j 1, then x Z(G) n. The following lemma seems well-known, but we could not find a reference and so we provide a proof here. Let S be a subset of a group G. We write S for the subgroup of G generated by S. Let G be a group with w G. We write χ w : G G, x w 1 xw for the conjugation map. Lemma Let w F (y 0,..., y n ) with n 0. Suppose that there is a positive integer k such that σ k j (w) = w for 0 j n 1. Then w lies in the subgroup generated by y 0 y 1 y n. In addition, if σ k 1w = w and n 1, then w = 1.

21 A BRAIDED SIMPLICIAL GROUP 21 Proof. The proof is given by induction on n. The assertion is trivial for n = 0. Let n = 1. We may assume that k = 2t is an even integer. Let x 0 = y 1 1 and let x 1 = y 0 y 1. Then F (y 0, y 1 ) = F (x 0, x 1 ). Since σ 0 (y 0 ) = y 1 and σ 0 (y 1 ) = y 1 1 y 0 y 1, we have σ 2 0 = χ x1 and so σ k 0 = χ t x 1 = χ x t 1. We can write w as a reduced word in F (y 0, y 1 ) = F (x 0 ) F (x 1 ). Then w = x n 1 0 x l 1 1 x n s 0 x ls 1, where n j 0 for 2 j s and l j = 0 for 1 j s 1. Suppose that w x 1. There are two cases: n 1 0 or n 1 = 0. If n 1 = 0, then x t 1w wx t 1. This contradicts the assumption that χ x t 1 (w) = w. Otherwise, n 1 = 0 and s > 1. Then w = x l 1 1 x n 2 0 w and x t+l 1 1 x n 2 0 w x l 1 1 x l 2 0 w x t 1. This contradicts the assumption that χ x t 1 (w) = w. Hence w x 1 = y 0 y 1. Now suppose that the assertion holds for n 1 with n > 1. Since F (y 0,..., y n ) = F (y 0,..., y n 1 ) F (y n ) is a free product, we can write w as a word w = y l 0 n w 1 y l 1 n w t y lt n,

22 22 JIE WU where w j 1 F (y 0,..., y n 1) and l j = 0 for 1 j t 1. Because σ j (y n ) = y n for j < n 1, we have σ k j (w i ) = w i for 1 i t and 0 j n 2. Let x = y 0 y 1 y n 1. By induction, we have w i x for 1 i t and so w x, y n. Let q : F (y 0, y 1,..., y n ) F (y n 1, y n ) be the projection defined by q(y j ) = 1 for j < n 1 and q(y j ) = y j for j n 1. Then q σ n 1 = σ n 1 q. Since σ n 1 w = w, we have σ n 1 (q(w)) = q(w) and so q(w) y n 1 y n. Because the restriction q x,yn : x, y n F (y n 1, y n ) is an isomorphism, we have w x y n = y 0 y 1 y n

23 A BRAIDED SIMPLICIAL GROUP 23 and hence the result. Let K n be the pure braided group, that is, K n is the kernel of the canonical epimorphism B n Σ n. Recall that the group B n is defined as the subgroup of B n+1 generated by σ i for i 0. There is a canonical isomorphism φ: B n B n with φ(σ i ) = σ i 1. We write K n for φ 1 (K n ). Recall that π n+1 (F (S 1 )) = Z(F (S 1 )) n+1 /B(F (S 1 )) n+1. Consider the actions of two groups B n+2 and B n+1 on F (S 1 ) n+1 /B(F (S 1 )) n+1. We have Theorem If n 2, then in F (S 1 ) n+1 /B(F (S 1 )) n+1, 1) the fixed set of the pure braided group K n+1-action is Z π n+1 (F (S 1 )); 2) the fixed set of the K n+2 -action is π n+1 (F (S 1 )). Proof. (1). Let y 1 = (y 0 y 1 y n ) 1. (1). We show that the homotopy fixed set of K n+1 on F (S 1 ) n+1 is generated by y 1 and Z(F (S 1 )) n+1. Assertion 1 will follow from this statement.

24 24 JIE WU Let w be a homotopy fixed point of the K n+1-action on F (S 1 ) n+1. Since σ 2 kd 0 = d 0 σ 2 k+1 for k 1, we have σ 2 kd 0 (w) = d 0 (w) for each k 1. By Lemma 2.13, we have d 0 (w) = 1. Now for each 1 j n + 1, we have σkd 2 j = d j σ 2 k+1 if j k + 1; d j σ 1 j 1 σ2 j 2 σ j 1 if j = k + 2; d j σ 2 k if j > k + 2. By Lemma 2.13, there exists integers k 1, k 2,..., k n+1 such that d j (w) = y k j 1 for 1 j n + 1. Since d k y 1 = y 1 for k > 0, we have y k j 1 = d j (y k j 1) = d j d j w = d j d j+1 w = d j (y k j+1 1 ) = y k j+1 1 for 1 j n and so k 1 = k 2 = = k n+1.

25 A BRAIDED SIMPLICIAL GROUP 25 Let w = y k 1 1 w. Then d j (w ) = 1 for each 0 j n+1 and w Z(F (S 1 )) n+1. This shows that w lies in the subgroup generated by y 1 and cycles Z(F (S 1 )) n+1. This finishes the proof of the statement and hence assertion 1 holds. (2). Let w be a homotopy fixed point of K n+2. By assertion 1, w lies in the subgroup generated y 1 and Z(F (S 1 )) n+1. Since π n+1 (F (S 1 )) = Z(F (S 1 )) n+1 /B(F (S 1 )) n+1 is the centre of F (S 1 ) n+1 /B(F (S 1 )) n+1, we have w y a 1w for some a Z and w Z(F (S 1 )) n+1. Recall that σ 1 (y 1 ) = y 0. By Lemma 2.8, we have σ 2 1(w ) w and so y a 1w w σ 2 1(w) σ 2 1(y a 1)w y 2a 0 w. Thus (y 0 y 1 y n 1 ) a = (d 1 (y 1 ) a = d 1 (y a 1w ) = d 1 (y 2a 0 w ) = 1. It follows that a = 0 and hence assertion 2 holds. Note: In F (S 1 ) n+1 /B n+1, since any element in the homotopy group is a homotopy fixed point of B n+2, the fixed set of B n+1 is Z π n+1 (F (S 1 )) and the fixed set of B n+2 is π n+1 (F (S 1 )).

26 26 JIE WU 2.4. Some Remarks on the simplicial group F (S n ). In the beginning of this section, we use a consequence of [16, Lemma 4.1] to get a simplicial group model for ΩS 2 by using Milnor s construction F (S 1 ). Below we provide a method to test certain simplicial free group models for ΩS n+1 by counting rank. The simplicial group F (S n ) has the following uniqueness property. Let G be a (simplicial) group. We write {Γ k G} k 1 for the descending central series of G starting with Γ 1 G = G. The completion of G with respect to the descending central series is the (simplicial) pro-group G = lim k G/Γ k (G). Let L(G) = k 1 Γ k (G)/Γ k+1 (G) be the (simplicial) Lie algebra induced by the (simplicial) group G. We write {Γ (p) k (G)} k 1 for the mod p descending central series of G. The induced (simplicial) restricted Lie algebra is denoted by L (p) (G) = k 1 Γ (p) k (G)/Γ(p) k+1 (G). The p-completion of G is the (simplicial) pro-group G p = lim k G/Γ (p) k (G). Proposition Let G be a simplicial group. Suppose that 1) G q = 1 for q < n and 2) G n+k is a free group of rank ( ) n+k n for k 0. Then there is a simplicial monomorphism φ: F (S n ) G such that

27 A BRAIDED SIMPLICIAL GROUP 27 1) φ is a simplicial monomorphism; 2) φ is a homotopy equivalence; 3) the map φ : F (S n ) G is an isomorphism; 4) the map φ : F (S n ) p G p is an isomorphism for any prime p. Proof. Let z be the generator for the group G n = Z and let fz : [n] G be the representing map of the element z, where [n] is the standard n-simplex. Since G n 1 = 1, we have d j z = 1 for each j and so the map f z factors through the n- sphere S n. Let g z : S n G be the resulting map. By the universal property of the functor F, there is a (unique) simplicial homomorpism φ: F (S n ) G such that φ S n = g z. Let φ ab : F (S n ) ab = K(Z, n) G ab be the induced map of the abelianization of simplicial groups. By assumption on G, the abelianization G ab is the minimal simplicial group K(Z, n) and so the map φ ab is an isomorphism. Since each G n is a free group, L(G) is the (simplicial) free Lie algebra generated by G ab. Thus (4) L(φ): L(F (S n )) L(G) is an isomorphism. It follows that φ is a monomorphism and φ : F (S n ) G is an isomorphism, which are assertions 1 and 3. By Equation (4), Curtis convergence theorem [7] holds for G and so φ is a homotopy equivalence by the Adams spectral

28 28 JIE WU sequence [8], which is assertion 2. Similarly, L (p) (φ): L (p) (F (S n )) L (p) (G) is an isomorphism and hence assertion 4. Corollary Let G be a simplicial group such that each G n is a free group of rank n. Then G is a simplicial group model for ΩS 2. Note. The point in Proposition 2.15 is that for any given simplicial free group G, we may just count the rank of each G n to see whether we obtain a simplicial group model for ΩS n+1. Note. By looking at Proposition 2.1, the braid group actions on F (S 1 ) are certain automorphisms that twist the faces and degeneracies. It seems interesting to have a generalization of these results on F (S n ) for general n. 3. The Geometry of the Group G(n) and the Proof of Theorem 1.3 In this section we give a proof of Theorem 1.3. First we need to represent the group G(n) as the fundamental group of a certain space X n. We will show that the braid group action on G(n) is induced by certain self homeomorphisms of X n The Geometry of the Group G(n). Let F n = F (x 1,..., x n ) and let ˆF n be the quotient group of F n by the single relation x 1 x 2 x n = 1. Let B n act on F n in

29 A BRAIDED SIMPLICIAL GROUP 29 Artin s sense (see Subsection 2.1 above) and let ˆB n be the image of B n in Aut( ˆF n ). We recall a geometric interpretation of the B n -action on F n and ˆF n. Let D 2 be the unit disc and let Q n = {q 1,..., q n } be a set of distinct fixed points of D 2 \ D 2. Then π 1 (D 2 \ Q n ) is a free group of rank n. Let x 1,..., x n be a basis for π 1 (D 2 \Q n ), where x i is represented by a simple loop which encloses the boundary point q i, but no boundary point q j for j i. According to [4, Theorem 1.10], the braid group B n is precisely the group of automorphisms of π 1 (D 2 \ Q n ) that are induced by self homeomorphisms of D 2 \ Q n that keep the boundary of D 2 fixed pointwise. (This result is due to Artin as well and can be regarded as a third definition of the group B n.) Let D 2 = S 2 S 2 be the canonical embedding of the lower hemisphere. Then π 1 (S 2 \ Q n ) = ˆF n and ˆB n is precisely the group of automorphisms of φ 1 (S 2 \ Q n ) that are induced by self homeomorphisms of S 2 \ Q n that keep the upper hemisphere S 2 \ S 2 fixed pointwise. Now let Q n,i = Q n \{q i } for each 1 i n. The n-fold diagonal map n : D 2 (D 2 ) n induces the map ψ n = n D 2 \Q n : D 2 \ Q n (D 2 \ Q n,1 )... (D 2 \ Q n,n ), x (x,..., x). It follows that there is a (unique) covering space Y n of n D 2 \ Q n,i such that i=1 n π 1 (Y n ) = Im(ψ n : π 1 (D 2 \ Q n ) π 1 ( D 2 \ Q n,i )) i=1

30 30 JIE WU n and the map ψ n lifts to Y n (uniquely). Let q n : Y n D 2 \ Q n,i be the covering map and let ψ n : D 2 Y n be the lifting of ψ n, that is q n ψ n = ψ n. Since the map ψ n is an embedding, so is ψ n. Observe that π 1 (Y n ) is the quotient group of F (x 1,..., x n ) modulo Relation (R 2 ) in the definition of the group G(n). Let X n = Y n S 2 \ Q n. D 2 \Q n By the Seifert-van Kampen theorem, we have i=1 Lemma 3.1. The fundamental group π 1 (X n ) is isomorphic to G(n). Note. The space Y n is a K(π, 1) and π 1 (Y n ) is isomorphic to F (S 1 ) n /NF (S 1 ) n, the quotient group of F (S 1 ) n modulo the Moore chains. On the other hand, X n is not a K(π, 1) in general because X n is an n-dimensional CW -complex and π 1 (X n ) = G(n) contains finite subgroups in general. According to [16], the maximal finite subgroup of G(n) is precisely the centre Z(G(n)) = π n (S 2 ) for n 4. Fred Cohen asked how to make X n into a K(π, 1) by considering the Borel construction on a certain group action on X n. Cohen s problem remains open. Let β be an element in the braid group B n. By the third definition of the group B n [4, Theorem 1.10], there is a homeomorphism h: D 2 \ Q n D 2 \ Q n such that h keeps the boundary S 1 of D 2 fixed pointwise and β = h : π 1 (D 2 \ Q n ) π 1 (D 2 \ Q n ).

31 A BRAIDED SIMPLICIAL GROUP 31 The self homeomorphism h of D 2 \Q n has a unique extension h to D 2 which permutes the points of Q n. Thus there exists a (unique) element µ β in Σ n such that the diagram D 2 \ Q n D 2 \ Q n,i = h D 2 \ Q n = di (h) = h D 2 \Q n,i D 2 \ Q n,µβ (i) commutes. We write d i (β) for the isomorphism d i (h) : π 1 (D 2 \ Q n,i ) π 1 (D 2 \ Q n,µβ (i)). As a geometric braid, d i (β) is the braid on n 1 strings obtained from β by deleting the i-th string. Note. β K n if and only if µ β = 1. In this case d i (h) is an self homeomorphism of D 2 \ Q n,i. Consider the commutative diagram D 2 \ Q ψn n = h D 2 \ Q n ψn n D 2 \ Q n,i i=1 n = i=1 d i(h) n D 2 \ Q n,i. i=1 i=1 i=1 n n The self homeomorphism d i (h) of D 2 \ Q n,i induces a unique self homeomorphism h of Y n such that h ψ n = ψ n h and q n h = ( n h i ) q n. Let h be an self i=1

32 32 JIE WU homeomorphism of S 2 \ Q n such that h D 2 \Q n = h and h S 2 \D2 is the identity map. Then the maps h and h define an self homeomorphism h = h h of X n. Clearly the map h : π 1 (X n ) π 1 (X n ) is the automorphism induced by β : π 1 (D 2 \ Q n ) π 1 (D 2 \ Q n ). Thus we have Proposition 3.2. The B n -action on G(n) is induced by certain self homeomorphisms of X n Proof of Theorem 1.3. Recall that the centre of B n is the infinite cyclic subgroup generated by (σ 1 σ 2 σ n 1 ) n for n 3. (See [4, Corollary 1.8.4].) Clearly the element (σ 1 σ n 1 ) n acts trivially on ˆF n. Thus ˆB n is a quotient group of B n /Z(B n ), where Z(G) denotes the centre of a group G. Lemma 3.3. Let n 3. Then the group ˆB n is isomorphic to B n /Z(B n ). Proof. By [4, Lemma ], the kernel of K n Aut( ˆF n ) is Z(K n ) = Z(B n ). Let G be the kernel of B n Aut( ˆF n ). By the commutative diagram Σ n Aut( ˆF n /Γ 2 ˆFn ) = GL(n 1, Z) B n Aut( ˆF n ),

33 we have G = Z(B n ) and hence the result. A BRAIDED SIMPLICIAL GROUP 33 Let C n = {β B n d i (β) = 1 for 1 i n}. Then C n is a normal subgroup of B n. Let δ i : F n = π 1 (D 2 \Q n ) F n,i = π 1 (D 2 \Q n,i ) be induced by the inclusion. Observe that the group F n,i is the free group generated by x 1,..., x i 1, x i+1,..., x n and the homomorphism δ i is given by δ i (x i ) = 1 and δ i (x j ) = x j for j i. There is commutative diagram F n δ i F n,i β F n d i (β) δ µβ (i) F n,µβ (i) for any β B n by the definition of d i (β). Lemma 3.4. The kernel of the representation B n Aut(π 1 (Y n )) is precisely the subgroup C n for each n. Proof. Consider the commutative diagram F n π 1 (Y n ) n i=1 F n,i β F n π 1 (Y n ) n F n,i. i=1 n i=1 d i(β)

34 34 JIE WU If β C n, then each d i (β) = 1. Thus C n Ker(B n Aut(π 1 (Y n ))) Conversely, suppose that β Ker(B n Aut(π 1 (Y n ))). By the commutative diagram F n β F n n i=1 π 1 (Y n ) d i(β) π1 (Y n ) π 1 (Y n ) F ni d i (β) F nµβ (i), we have d i (β) = 1 for all i and so β C n. This finishes the proof. Consider the map ψ n : S 2 \ Q n n S 2 \ Q n,i, i=1, x (x,..., x). n There is a (unique) covering space Z n of S 2 \ Q n,i such that the map ψ n lifts to Z n and π 1 (Z n ) is given by the image of ψ n. i=1 Note. The space Z n is a K(π, 1) and π 1 (Z n ) is isomorphic to F (S 1 ) n 1 /Z(F (S 1 )) n 1, the quotient group of F (S 1 ) n 1 modulo the cycles.

35 A BRAIDED SIMPLICIAL GROUP 35 Let ˆF n,i = π 1 (S 2 \ Q n,i ) and let ˆδ i : ˆFn ˆF n,i be induced by the inclusion S 2 \ Q n S 2 \ Q n,i. Let h be an self homeomorphism of D 2 \Q n that keeps the boundary of D 2 fixed pointwise. We write ˆd i (h) for the homeomorphism S 2 \ Q n,i S 2 \ Q n,µh (i) induced by the canonical extension of h to S 2. Let β B n be represented by h. We write ˆd i (β) for the isomorphism ˆF n,i ˆF n,µβ (i) induced by the map ˆd i (h). Then there is a commutative diagram ˆF n ˆδi ˆFn,i β ˆF n ˆδµβ (i) ˆd i (β) ˆFn,µβ (i) and so the B n -action on ˆF n induces a B n -action on π 1 (Z n ). Let D n = {β B n ˆd i (β) = 1 for 1 i n}. Clearly D n is a normal subgroup of B n. By the proof of Lemma 3.4, we have Lemma 3.5. The kernel of the representation B n Aut(π 1 (Z n )) is precisely the subgroup D n. Proof of Theorem 1.3. The canonical inclusions of Y n and S n \ Q n into Z n induce an inclusion of X n into Z n. Let D n be the kernel of the representation B n Aut(G(n)) = Aut(π 1 (X n )).

36 36 JIE WU By Lemmas , we have Z C n = Z(Bn ) C n D n D n. It suffices to show that D n Z(B n ) C n. First we show that K n D n Z(B n ) C n. Suppose that β K n D n. Let x 0 = x 1... x n F n and let α = (σ 1 σ n 1 ) n be the generator for Z(B n ). Then α = χ x0 : F n F n, where χ w (x) = w 1 xw is the conjugation map. Observe that d i (β): F n,i F n,i is a braid for each i, that is, d i (β) lies in the braid group B n 1 as a subgroup of Aut(F n,i ). Since ˆd i (β): ˆFn,i ˆF n,i is the identity for each i, the element d i (β) lies in the kernel of B n 1 Aut( ˆF n,i ) and so there exists an integer l i such that d i (β) = χ (δi (x 0 ) l i for each i. For each 1 i < j n, there exists a unique automorphism d ij (β ij ): F n,ij F n,ij such that the diagram F n δ ij F n,ij β F n δ ij F n,ij d ij (β) commutes, where F n,ij = F (x 1,..., ˆx i,..., ˆx j,..., x n ) and the homomorphism δ ij is given by δ ij (x k ) = x k for k i, j and δ ij (x i ) = δ ij (x k ) = 1. It follows that χ δij(x0 ) l i = d j d i (β) = d ij (β) δ ij = d i d j (β) = χ δij (x 0 ) l j.

37 A BRAIDED SIMPLICIAL GROUP 37 Since F n,ij is a free group of rank at least 2, F n,ij embeds into Aut(F n,ij ) as inner automorphisms and so l i = l j for i < j. Thus βα l 1 C n. In other words, K n D n Z(B n ) C n. By the commutative diagram ˆB n Σ n Aut(π 1 (Z n )) Aut(π 1 (Z n )/Γ 2 π 1 (Z n )) = GL(n 1; Z), we have D n = K n D n and hence the result. References [1] E. Artin, Theorie der Zopfe, Hamburg Abh. 4 (1925), [2] E. Artin, Theory of braids, Ann. of Math. 48 (1947), [3] A. J. Berrick, Y. L. Wong and J. Wu, On Brunian braids and the homotopy groups, preprint. [4] J. S. Birman, Braids, links, and mapping class groups, Annals of Math. Studies 82 Princeton University Press, (1975). [5] F. R. Cohen, Cohomology of braid groups, Bull. Amer. Math. Soc. 79 (1973), [6] F. Cohen and J. Wu, A representation of F (S 1 ) into braids, preprint. [7] E. B. Curtis, Some relations between homotopy and homology, Ann. of Math. 83 (1965), [8] E. B. Curtis, Simplicial homotopy theory, Advances in Math. (1971),

38 38 JIE WU [9] E. B. Curtis and M. Mahowald, The unstable Adams spectral sequence for S 3, Contemp. Math., 96 (1989), [10] Z. Feidorowicz and J-L. Loday, Crossed simplicial groups and their associated homology, Trans. AMS 326 (1991), [11] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Interscience Publishers (1966). [12] J. P. May, Simplicial objects in Algebraic Topology, Math Studies 11, van Nostrand (1967). [13] J. Milnor, On the construction F (K), Algebraic Topology- A Student Guide, by J.F. Adams, , Cambridge Univ. Press. [14] J. C. Moore, Homotopie des complexes monöideaux, Seminaire Henri Cartan ( ). [15] H. Toda, Composition methods in homotopy groups of spheres, Princeton Univ. Press, [16] Jie Wu, Combinatorial descriptions of the homotopy groups of certain spaces, Math. Proc. Camb. Philos. Soc. to appear. Department of Mathematics, National University of Singapore, Singapore , Republic of Singapore, matwuj@nus.edu.sg