Mirror Reflections on Braids and the Higher Homotopy Groups of the 2-sphere A gift to Professor Jiang Bo Jü Jie Wu Department of Mathematics National University of Singapore www.math.nus.edu.sg/ matwujie June 21, 2007
Artin Braids and Homotopy Groups Homotopy Groups Braid Groups Main Results Methods of the Proofs Remarks
Homotopy Groups Let X be a topological space. The homotopy groups are defined by π n (X) = [S n, X], the set of the (pointed) homotopy classes of (pointed) continuous maps from the n-sphere S n to X. π 0 (X) is the set of the path-connected components of X. π 1 (X) is the fundamental group of X. π n (X) is abelian for n 2. Fundamental Problem in Algebraic Topology: Determine the (general) homotopy groups of spheres. Some connections between the (general) homotopy groups of S 2 and the braid groups are given in this talk.
Braid Groups The braid group B n is generated by σ 1, σ 2,..., σ n 1 subject to the braiding relations: σ i σ j = σ j σ i for i j 2, and σ i σ i+1 σ i = σ i+1 σ i σ i+1 for each i. 1 i-1 i i+1 n σ i
Pure Braid Groups The symmetric group S n is the quotient group of B n subject to the following additional relations: σ 2 i = 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 for 1 i < j n. i j
Brunnian Braids 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 B n which is denoted by Brun n. (σ 1 σ 2-1 ) 3 is Brunnian
Theorem The Brunnian group Brun n is the normal subgroup of the pure braid group 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. Note. As a group, Brun n is a free group of rank for n 3.
The Automorphism θ on P n 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, 1 z 2 z 1, 1 z 3 z 1,..., 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). ).
The Automorphism θ on P 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.
The Automorphism θ on P n The action of θ on P n is given by for 2 i < j n, and where θ(a i,j ) = A i,j θ(a 1,j ) = A 1 1,j A 0,jA 1,j, 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 1 σ n 2 σ j ) 1 (σ j 1 σ 2 σ 2 1 σ 1 σ j 1 ) 1 is an element in P n for 1 j n. 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 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.
Boundary Brunnian Braids 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 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 ).
Description of Boundary Brunnian Braids 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 Meaning of boundary Brunnian braids 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. Theorem: the group Bd n is the commutator subgroup in B n of Brun n and the kernel of q. Theorem: 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.
Questions Is the group P n / Bd n torsion free? 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 β Bd 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 ).
Theorem 1 Denote by Z (G) the center of a group G. Let n 4. (Bd n, P n ) = Brun n θ(brun n ) with an isomorphism of groups (Bdn, P n )/ Bd n = πn (S 2 ). There are isomorphisms of groups Z (P n / Bd n ) = π n (S 2 ) Z, Z (B n / Bd n ) = {α π n (S 2 ) 2α = 0} Z.
Mirror Reflection Problem 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 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 ).
Methods of the Proofs simplicial and -structure on the sequence of braid groups. The conjugation action of B n on pure braid group P n. The Hopf Invariants in homotopy theory have to be used for proving Theorem 2. Thus the method is really a combination of simplicial method, braid groups and homotopy theory.
Homotopy Groups of S 2 It was known by Serre 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 that 4 Tor 2 (π n (S 2 )) = 0. Namely Tor 2 (π n (S 2 )) is a finite direct sum of Z/2 and Z/4. Furthermore, a result of Selick 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 π n (S 2 ) = Fix χ (B n / Bd n ) = (Bd n, P n )/ Bd n = Tor(Z (P n / Bd n )) and {α π n (S 2 ) 2α = 0} = Tor(Z (B n / Bd n )).
Historical Remarks As a combinatorial tool for studying homotopy theory, simplicial groups were first studied by J. C. Moore. 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. Milnor 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.
Historical Remarks It is possible that two simplicial groups with the same homotopy type have sharply different group structures. 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. 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. For this model, the descending central series will not give any information as the groups are perfect, but word filtration provides different information.
Historical Remarks 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 one of my papers (Math. Proc. Camb. Phi. Soc. 2001), 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. 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 now gives a positive answer to this question.
Historical Remarks A connection between the braid groups and the general homotopy groups of S 2 was found in one of my papers (Proc. London Math. Soc.), 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. Further development is given in the joint paper with F. Cohen, J. Berrick and Y. L. Wong (Journal of AMS 2006), where 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 joint papers with F. Cohen.
Historical Remarks 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.
Remarks 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. A theorem in this article states that the 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.
Remarks Notice that the conjugation action of B n on Brun n induces a B n -module structure on Brun ab n. By a theorem in this article, 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) 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. Reference of this Talk: Jie Wu, Artin braid groups and homotopy groups, preprint, available through my web site. The End.