Simplicial Objects and Homotopy Groups

Simplicial Objects and Homotopy Groups Jie Wu Department of Mathematics National University of Singapore July 8, 2007

Simplicial Objects and Homotopy Groups -Objects and Homology Simplicial Sets and Homotopy Simplicial Groups Braids and Homotopy Groups

Definitions A -set means a sequence of sets X = {X n } n 0 with faces d i : X n X n 1, 0 i n, such that for i j, which is called the -identity. d i d j = d j d i+1 (1) One can use coordinate projections for catching -identity: d i : (x 0,..., x n ) (x 0,..., x i 1, x i+1,..., x n ). A -map f : X Y means a sequence of functions f : X n Y n for each n 0 such that f d i = d i f. A -set G = {G n } n 0 is called a -group if each G n is a group, and each face d i is a group homomorphism.

-set is a contravariant functor Let O + be the category whose objects are finite ordered sets and whose morphisms are functions f : X Y such that f (x) < f (y) if x < y. The objects in O + are given by [n] = {0, 1,..., n} for n 0 and the morphisms in O + are generated by d i : [n 1] [n] the ordered embedding missing i. Proposition. Let S denote the category of sets. -sets are one-to-one correspondent to contravariant functors from O + to S. a -group means a contravariant functor from O + to the category of groups. More abstractly, for any category C, a -object over C means a contravariant functor from O + to C. In other words, a -object over C means a sequence of objects over C, X = {X n } n 0 with faces d i : X n X n 1 as morphisms in C.

Example of -sets The n-simplex + [n], as a -set, is as follows: + [n] k = {(i 0, i 1,..., i k ) 0 i 0 < i 1 < < i k n} for k n and + [n] k = for k > n. The face d j : + [n] k + [n] k 1 is given by d j (i 0, i 1,..., i k ) = (i 0, i 1,..., îj,..., i k ), that is deleting i j. Let σ n = (0, 1,..., n). Then any elements in [n] can be written an iterated face of σ n. Proposition. Let X be a -set and let x X n be an element. Then there exists a unique -map, called the representing map of x, such that f x (σ n ) = x. f x : + [n] X

Abstract Simplicial Complexes An abstract simplicial complex K is a collection of finite nonempty sets, such that if A is an element in K, so is every nonempty subset of A. The element A of K is called a simplex of K; its dimension is one less than the number of its elements. Each nonempty subset of A is called a face of A. Let K be an abstract simplicial complex with vertices ordered. Let K n be the set of n-simplices of K. Define the faces d i : K n K n 1, 0 i n, as follows. If {a 0, a 1,..., a n } is an n-simplex of K with a 0 < a 1 < < a n, then define d i {a 0, a 1,..., a n } = {a 0, a 1,..., a i 1, a i+1,..., a n }. Proposition. Let K = {K n } n 0 with faces defined as above. Then K is a -set.

Polyhedral A -set X is called polyhedral if there exists an abstract simplicial complex K such that X = K. Theorem. Let X be a -set. Then X is polyhedral if and only if the following holds: 1. There exists an order of X 0 such that, for each x X n, f x (0) f x (1) f x (n). 2. The function φ: n 0 X n 2 X 0 is one-to-one.

-complex The standard geometric n-simplex n is defined by n n = {(t 0, t 1,..., t n ) t i 0 and t i = 1}. Define d i : n 1 n by setting i=0 d i (t 0, t 1,..., t n 1 ) = (t 0,..., t i 1, 0, t i,..., t n 1 ). A -complex structure on a space X is a collection of maps C(X) = {σ α : n X α J n n 0}. such that 1. σ α Int( n ) : Int( n ) X is injective, and each point of X is in the image of exactly one such restriction σ α Int( n ). 2. For each σ α C(X), each face σ α d i C(X). 3. A set A X is open if and only if σ 1 α (A) is open in n for each σ α C(X).

-set from -complex Let X be a -complex. Define C n (X) = {σ α : n X α J n } C(X) with d i : Cn (X) Cn 1 (X) given by for 0 i n. d i (σ α ) = σ α d i C (X) = {C n (X)} n 0 is a -set.

Geometric Realization of -sets Let K be a -set. The geometric realization K of K is defined to be K = x K n n 0 ( n, x)/ = n K n /, n=0 where ( n, x) is n labeled by x K n and is generated by (z, d i x) (d i z, x) for any x K n and z n 1 labeled by d i x. For any x K n, let σ x : n = ( n, x) K be the canonical characteristic map. The topology on K is defined by A K is open if and only if the pre-image σx 1 (A) is open in n for any x K n and n 0. Proposition. Let K be a -set. Then K is -complex.

Homology of -sets a chain complex of groups means a sequence C = {C n } of groups with differential n : C n C n 1 such that n n+1 is trivial, that is Im( n+1 ) Ker( n ) and so the homology is defined by H n (C) = Ker( n )/Im( n+1 ), which is a coset in general. Proposition. Let G be a -abelian group. Define n n = ( 1) i d i : G n G n 1. i=0 Then n 1 n = 0, that is, G is a chain complex under. Let X be a -set. The homology H (X; G) of X with coefficients in an abelian group G is defined by H (X; G) = H (Z(X) G, ), where Z(X) = {Z(X n )} n 0 and Z(X n ) is the free abelian group generated by X n.

Simplicial and Singular Homology Let X be a -complex. Then simplicial homology of X with coefficients in an abelian group G is defined by For any space X, define H (X; G) = H (C (X); G). S n (X) = Map( n, X) be the set of all continuous maps from n to X with d i = d i : S n (X) = Map( n, X) S n 1 (X) = Map( n 1, X). for 0 i n. Then S (X) = {S n (X)} n 0 is a -set. The singular homology H (X; G) = H (S (X); G).

Definition A simplicial set means a -set X together with a collection of degeneracies s i : X n X n+1, 0 i n, such that for j < i, for j i and d j d i = d i 1 d j (2) s j s i = s i+1 s j (3) s i 1 d j j < i d j s i = id j = i, i + 1 s i d j 1 j > i + 1. The three identities for d i d j, s j s i and d i s j are called the simplicial identities. A simplicial map f means a sequence of functions f : X n Y n such that d i f = fd i and s i f = fs i. One can use deleting-doubling coordinates for catching simplicial identities. (4)

Simplicial Set is a Functor Let O be the category whose objects are finite ordered sets and whose morphisms are functions f : X Y such that f (x) f (y) if x < y. The objects in O are given by [n] = {0,..., n} for n 0, which are the same as the objects in O +. The morphisms in O are generated by d i, which is defined in O +, and the following morphism ( s i : [n+1] [n] s i 0 1 i 1 i i + 1 i + 2 = 0 1 i 1 i i i + 1 for 0 i n, that is, s i hits i twice. Exercise. simplicial sets are one-to-one correspondent to contravariant functors from O to S. Let C be a category. A simplicial object over C means a contravariant functor from O to C, i.e. a sequence of objects X = {X n } n 0 with face morphisms d i : X n X n 1 and degeneracy morphisms s i : X n X n+1, 0 i n, such that the three simplicial identities hold.

Geometric Realization Let X be a simplicial set. Then its geometric realization X is a CW -complex defined by X = ( n, x)/ = n X n /, x X n n 0 n=0 where ( n, x) is n labeled by x X n and is generated by (z, d i x) (d i z, x) for any x X n and z n 1 labeled by d i x, and (z, s i x) (s i z, x) for any x X n and z n+1 labeled by s i x. Note that the points in ( n+1, s i x) and ( n 1, d i x) are identified with the points in ( n, x). X is a CW -complex.

n-simplex The n-simplex [n], as a simplicial set, is as follows: [n] k = {(i 0, i 1,..., i k ) 0 i 0 i 1 i k n} for k n. The face d j : [n] k [n] k 1 is given by d j (i 0, i 1,..., i k ) = (i 0, i 1,..., îj,..., i k ), that is deleting i j. The degeneracy s j : [n] k [n] k+1 is defined by s j (i 0, i 1,..., i k ) = (i 0, i 1,..., i j, i j,..., i k ), that is doubling i j. Let σ n = (0, 1,..., n) [n] n. Then any elements in [n] can be written as iterated compositions of faces and degeneracies of σ n. Let X be a simplicial set and let x X n be an element. Then there exists a unique simplicial map, called representing map of x, f x : [n] X

Basic Constructions Cartesian Product: (X Y ) n = X n Y n. X Y = X Y under compact generated topology. The geometric realization of a simplicial group is a topological group (under compactly generated topology). Wedge: (X Y ) n = X n Y n, union at the basepoint. X Y = X Y. Smash Product: X Y = (X Y )/(X Y ). X Y = X Y under compactly generated topology.

Mapping Space Let σ n = (0, 1,..., n) [n] be the nondegenerate element. d i = f di σ n : [n 1] [n] s i = f si σ n : [n + 1] [n] Let Map(X, Y ) n = Hom S (X [n], Y ) and let d i = (id X d i ) : Map(X, Y ) n = Hom S (X [n], Y ) Hom S (X [n 1], Y ) = Map(X, Y ) n 1, s i = (id X s i ) : Map(X, Y ) n = Hom S (X [n], Y ) Hom S (X [n + 1], Y ) = Map(X, Y ) n+1 for 0 i n. Map(X, Y ) = {Map(X, Y ) n } n 0 with d i and s i is a simplicial set, which is called the mapping space from X to Y.

Homotopy Let I = [1]. As a simplicial set, n+1 i {}}{{}}{ I n = {t i = ( 0,..., 0, 1,..., 1) 0 i n + 1} n+1 Given a simplicial set X, the simplicial subsets X 0 and X 1 of X I are given by (X 0) n = {(x, t 0 ) x X n }, (X 1) n = {(x, t 1 ) x X n }. Let f, g : X Y be simplicial maps. We call f homotopic to g if there is a simplicial map F : X I Y such that F X 0 = f and F X 1 = g in which case write f g. If A is a simplicial subset of X and f, g : X Y are simplicial maps such that f A = g A, we callf homotopic to g relative to A, denoted by f g rel A, if there is a homotopy F : X I Y such that F X 0 = f, F X 1 = g and F A I = f. i

Fibrant Simplicial Set Let X be a simplicial 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 j k and k, j + 1 i. A simplicial set X is called fibrant (or Kan complex) 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. Theorem. Let X and Y be simplicial sets. If Y is fibrant, then [X, Y ] = Hom S (X, Y )/ is the same as [ X, Y ].

Simplicial Fibration A simplicial map p : E B is called a (Kan) fibration if for every commutative diagram of simplicial maps Λ k [n] i E p [n] B there is a simplicial map θ : [n] X (the dotted arrow) making the diagram commute, where i is the inclusion of Λ k [n] in [n]. Let v B 0 be a vertex. Then F = p 1 (v), that is F = {p 1 (s n v} n 0 is called a fibre of p over v. Let p : E B be a fibration and let F = p be the fibre. Suppose that E or B is fibrant. Then there is a long exact sequence on homotopy groups.

Moore Paths and Moore Loops Let X be a pointed simplicial set. The Moore Path is defined by setting (PX) n = {x X n+1 d 1 d 2 d n+1 x = } with faces di P = di+1 X (PX) n : (PX) n (PX) n 1 and degeneracies si P = si+1 X (PX) n : (PX) n (PX) n+1 for 0 i n. Proposition. PX is contractible. If X is fibrant, PX X is a fibration. Let X be a pointed fibrant simplicial set. The Moore loop ΩX is defined to be the fibre of p X : PX X. More precisely, (ΩX) n = {x X n+1 d 1 d 2 d n+1 x = and d 0 x = } with faces di Ω 0 i n. = d X i+1 and degeneracies sω i = s X i+1 for

Moore Postnikov System Let X be a simplicial set. For each n 0, define the equivalence relation n on X as follows: For x, y X q, we call x n y if each iterated face of x of dimension n agrees with the correspondent iterated face of y. Define P n X = X/ n for each n 0. Then we have the tower X P n X P n 1 X P 0 X called Moore-Postnikov system of X or coskeleton filtration of X.

Theorem Let X be a pointed fibrant simplicial set and let be the projection. Then p n : X P n (X) each p n : X P n X is a fibration, each P n X is fibrant, and for each n m, the projection P n X P m X is a fibration. p n : π q (X) π q (P n X) is an isomorphism for q n. Let q > n and let x be a spherical element in (P n X) q. Then x =. In particular, π q (P n X) = 0 for q > n. Let E n X be the fibre of the projection P n X P n 1 X. Then E n X is an Eilenberg-MacLane complex of K (π n (X), n).

Moore Chain Complex Let G be a simplicial group. Define N n G = n Ker(d j : G n G n 1 ). j=1 Then NG = {N n G, d 0 } is a chain complex. Moore Theorem. H n (NG) = π n (G) = π n ( G ). Moore cycles: Z n G = n j=0 Ker(d j : G n G n 1 ). Moore boundaries: B n G = d 0 (N n+1 G). π n ( G ) = Z n G/B n G.

Abelian Simplicial Groups Let G be a simplicial abelian groups. Define by setting n : G n G n 1 n (x) = n ( 1) j d j x. j=0 As we have seen for abelian -groups, (G, ) becomes a chain complex. Thus for a simplicial abelian group G we have two chain complexes (NG, d 0 ) and (G, ). Let x N n G. Then (x) = d 0 x because d j x = 0 for j > 0, and so (NG, d 0 ) is a chain subcomplex of (G, ). Theorem. Let G be a simplicial abelian group. Then the inclusion (NG, d 0 ) (G, ) induces an isomorphism on homology H (NG, d 0 ) = H (G, ).

Hurewicz Theorem Let X be a simplicial set. Let Z(X) = {Z(X n )} n 0 be the sequence of the free abelian group generated by X n. Then Z(X) is a simplicial abelian group. The integral homology of X is defined by H (X) = π (Z(X)) = H (Z(X), ). Hurewicz homomorphism The map j : X Z(X), x x induces a group homomorphism h n = j : π n (X) π n (Z(X)) = H n (X) for any fibrant simplicial set X and n 1. Let X be a fibrant simplicial set with π i (X) = 0 for i < n with n 2. Then H i (X) = 0 for i < n and is an isomorphism. h n : π n (X) H n (X)

Group Homology Let G be a monoid. Let WG be the simplicial set given by (WG) n = {(g 0, g 1,..., g n ) g i G} = G n+1 with faces and degeneracies given by d i (g 0,..., g n ) = { (..., gi 1, g i g i+1, g i+2,...) if i < n (g 0,..., g n 1 ) if i = n s i (g 0,..., g n ) = (g 0, g 1,..., g i, e, g i+1..., g n ). Let R be a commutative ring and let M be an R(G)-module. Then we have the simplicial abelian group R(WG) R(G) M. The homology of G with coefficients in M is then defined by H (G; M) = π (R(WG) R(G) M) = H (R(WG) R(G) M, ).

Milnor s Construction A pointed set S means a set S with a basepoint. Denote by F[S] the free group generated by S subject to the relation = 1. Let X be a pointed simplicial set. Milnor s construction is the simplicial group F[X], where F[X] n = F[X n ] with face and degeneracy homomorphisms induced by the faces and degeneracies of X. Theorem. F[X] ΩΣ X. The loop space of the suspension of X. James Construction. J(X) is the free monoid generated by X subject to the relation = 1. Theorem. If X is path-connected, then the inclusion J(X) F[X] is a homotopy equivalence.

Kan s Construction For a reduced simplicial set X, let GX be the simplicial group defined by 1. (GX) n is the quotient group of X n+1 subject to the relations: s 0 x = 1 for every x X n. Thus, as a group, (GX) n is the free group generated by X n+1 s 0 (X n ); or (GX) n = F[X n+1 /s 0 (X n )]. 2. The face and degeneracy operators are the group homomorphisms such that for x X n+1. d GX 0 x = (d 1 x)(d 0 x) 1, d GX i x = d i+1 x for i > 0, s GX i x = s i+1 x Theorem. GX Ω X.

Central Extension Theorem Proposition. The action of G 0 on G by conjugation induces an action of π 0 (G) on π n (G) for each n 0. We call a simplicial group G n-simple if π 0 (G) acts trivially on π n (G). A simplicial group G is called simple if it is n-simple for every n. Theorem. Let G be a simplicial group and let n 0. Suppose that G is n-simple. Then the homotopy group π n (G) is contained in the center of G n /B n G. Theorem. Let G be a reduced simplicial group such that G n is a free group for each n. Then there exits a unique integer γ G > 0 such that G n = {1} for n < γ G and rank(g n ) 2 for n > γ G. Furthermore, for n γ G + 1. π n (G) = Z (G n /B n G)

Abelianization of Moore Chains Let G be a group. Write G ab = G/[G, G] for the abelianization of the group G. Now given a simplicial group G, we have a chain complex of (non-abelian in general) groups (NG, d 0 ). Take the abelianization (N n G) ab of N n G for each n, the differential d 0 : N n G N n 1 G induces a homomorphism d0 ab : (N ng) ab (N n 1 G) ab and so a chain complex of abelian groups ((NG) ab, d0 ab). Theorem. Let G be a simplicial group such that (1) each G n is a free group and (2) π 0 (G) acts trivially on each π n (G). Then there is a decomposition H n ((NG) ab ) = π n (G) A n, where A n is a free abelian group. In particular, Tor(H ((NG) ab )) = Tor(π (G)).

Simplicial Group F [S 1 ] ΩS 2 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 = {s2 0, s 0σ, s 1 σ}, S3 1 = {s3 0, s 2s 1 σ, s 2 s 0 σ, s 1 s 0 σ}, and in general Sn+1 1 = {sn+1 0, x 0,..., x n }, where x j = s n ŝ j s 0 σ. The face d i : S 1 n+1 = {, x 0,..., x n } S 1 n = {, x 0,..., x n 1 } is given by d i s n+1 0 = s0 n and s0 n if j = i = 0 or i = j + 1 = n + 1 d i x j = x j if j < i x j 1 if j i. Similarly, s i x j = s i s n ŝ j s 0 σ = { xj if j < i x j+1 if j i.

Simplicial Group F [S 1 ] ΩS 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. Let y 0 = x 0 x 1 1,..., y n 1 = x n 1 xn 1 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 d i y j (0 i n + 1, 1 j n) given by d i y j = d i (x j x 1 j+1 ) = y j if j < i 1 1 if j = i 1 y j 1 if j i, y j if j < i 1 s i y j = y j y j+1 if j = i 1 y j+1 if j i, where y 1 = (y 0 y 1 y n 1 ) 1 and in this formula x n+1 = 1.

Simplicial Group F [S 1 ] ΩS 2 To describe all faces d i systematically in terms of projections, consider the free group of rank n in the following way. Let ˆF n+1 be the quotient of the free group F(z 0, z 1,..., z n ) subject to the single relation z 0 z 1 z n = 1. Let ẑ j be the image of z j in ˆF n+1. The group ˆF n+1 is written ˆF(ẑ 0, ẑ 1,..., ẑ n ) in case the generators ẑ j are used. Clearly ˆF n+1 = F(ẑ0, ẑ 1,..., ẑ n 1 ) is a free group of rank n. ˆF n+1 = π 1 (S 2 Q n+1 ). Define the faces d i and degeneracies s i on {ˆF n+1 } n 0 as follows: d i ẑ j = ẑ j if j < i 1 if j = i ẑ j 1 if j > i, s i ẑ j = ẑ j if j < i ẑ i ẑ i+1 if j = i ẑ j+1 if j > i. ˆF = {ˆFn+1 } n 0 with d i and s i defined as above is isomorphic to F[S 1 ].

Combinatorial Description of π (S 2 ) The Group G(n): generated by x 1, x 2,..., x n subject to the relations: 1) the ordered product of the generators x 1 x 2 x n = 1 and 2) the iterated commutators on the generators with the property that each generator occurs at least once in the commutator bracket. Theorem. For each n, the homotopy group π n (S 2 ) is isomorphic to the center of G(n). Fixed-point Set Theorem. The Artin representation induces an action of the braid group B n on G(n). Moreover the homotopy group π n (S 2 ) is isomorphic to the fixed set of the pure braid group action on G(n).

Example 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: [n 1] d i σ [n] d i (σ) [n 1] [n + 1] d i s i σ σ [n] [n] s i (σ) [n + 1] s i σ [n]

Crossed Simplicial Groups 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 stands are counted from initial points. 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.

Simplicial Structure on Configurations Let (M, d) be a metric space with basepoint w and let R + denote [0, ). A steady flow over M is a (continuous) map θ : R + M M such that (1) for any x M, θ(0, x) = x and for t > 0 0 < d(θ(t, x), x) t; (2) θ R+ {w} : R + {w} M is one-to-one; (3) there exists a function ɛ: R + (0, + ), t ɛ t, such that θ([0, ɛ t ) {θ(t, w)}) θ([t, ) {w}) for any t R +. Let M be a metric space with a steady flow. Then the sequence of groups B(M) π 1 = {π 1 (B(M, n + 1))} n 0 is a crossed simplicial group. Let M be a metric space with a steady flow. Then the sequence of groups F(M) π 1 = {π 1 (F(M, n + 1))} n 0 is a simplicial group.

-groups 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 a group homomorphism. The Moore complex NG = {N n G} n 0 of a -group G is defined by N n G = n Ker(d i : G n G n 1 ). i=1 Let G be a -group. Then d 0 (N n G) N n 1 G and NG with d 0 is a chain complex of groups. Let G be a -group. An element in B n G = d 0 (N n+1 G) is called a Moore boundary and an element in 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 π n (G) = H n (NG) = Z n G/B n G.

-groups Induced by Configurations A crossed -group is a -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 -map and (ii) for 0 i n, d i (gg ) = d i (g)d i µg (g ). A space M is said to have a good basepoint w 0 if there is a continuous injection θ : R + M with θ(0) = w 0. Let M be a space with a good basepoint. Then (1) B(M) π 1 is a crossed -group, (2) F(M) π 1 is a -group,

Brunnian Braids An element in π 1 (B(M, n)) is called a braid of n strings over M. A pure braid of n strings over M means an element in π 1 (F(M, n)). A braid is called Brunnian if it becomes a trivial braid when any one of its strings is removed. In the terminology of -groups, a braid β π 1 (B(M, n + 1)) = B(M) π 1 n is Brunnian if and only if d i β = 1 for 0 i n. In other words, the Brunnian braids over M are the Moore cycles in the -group B(M) π 1. The group of Brunnian braids of n strings over M is denoted by Brun n (M). Proposition. Let β be a Brunnian braid of n strings over a space M with a good basepoint. If n 3, then β is a pure braid.

Theorems There is an exact sequence of groups 1 Brun n+1 (S 2 ) Brun n (D 2 ) f Brun n (S 2 ) π n 1 (S 2 ) 1 for n 5, where f is induced from the canonical embedding f : D 2 S 2. Let F(S 2 ) π 1 be the -group defined above. Then, for each n 1, the Moore homotopy group π n (F(S 2 ) π 1) is a group, and there is an isomorphism of groups for n 4. π n (F(S 2 ) π 1 ) = π n (S 2 )

Theorem Let AP = {P n+1 } be the sequence of classical Artin pure braid groups with the simplicial structure given by deleting/doubling braids. Then AP is a reduced simplicial group because AP 0 = P 1 = 1. Since AP 1 = P 2 = Z, there is a unique simplicial homomorphism Θ: F[S 1 ] AP that sends the non-degenerate 1-simplex of S 1 to the generator of AP 1 = Z, where F[S 1 ] is Milnor s F[K ]-construction on S 1. The morphism of simplicial groups Θ : F[S 1 ] AP is an embedding. Hence the homotopy groups of F[S 1 ] are natural sub-quotients of AP, and the geometric realization of quotient simplicial set AP /F[S 1 ] is homotopy equivalent to the 2-sphere. Furthermore, the image of Θ is the smallest simplicial subgroup of AP which contains A 1,2.

The -group P Let δ : F(C, n + 1) F(C, n) be the map defined by ( ) 1 1 1 δ(z 1, z 2,..., z n ) =,,...,, 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 0. Then δ induces a group homomorphism : P n+1 P n. Proposition. Let χ: B n B n be the mirror reflection, that is, χ is an automorphism of B n such that χ(σ i ) = σ 1 i for all i. Let = χ : P n+1 P n. Then P = {P n } n 0 is -group with d 0 = and d 1,..., d n : P n P n 1 are given by deleting strands. Theorem. The -group P is fibrant, and there is an isomorphism π n (P) = π n (S 2 ) for every n.

Boundary Brunnian Braids The boundary Brunnian braid is defined to be Bd n = B n P = (Brun n+1 )) = (Brun n+1 )) as the subgroup of P n. Proposition. Bd n is a normal subgroup of B n. Theorem. The quotient groups P n /Bd n and B n /Bd n are finitely presented. Questions: Two 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.

Theorem 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 (Bdn, P n )/Bd n is the set of torsion elements in P n /Bd n. Let n 4. 1) (Bd n, P n )/Bd n = πn (S 2 ). 2) There are isomorphisms of groups Center(P n /Bd n ) = π n (S 2 ) Z Center(B n /Bd n ) = {α π n (S 2 ) 2α = 0} Z. This result gives a connection between the general higher homotopy groups of S 2 and the (special cases) of the conjugation problem on braids.

Mirror Reflection Theorem By moving our steps to the next, the mirror reflection χ: B n B n is a canonical operation on braids. Given a subgroup G of B n, one may ask whether there is a mirror symmetric braid β subject to G, that is, the braids β satisfying the equation of cosets χ(β)g = βg. The answer to this question 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-point of an automorphism φ of a group G. Theorem. 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 )