Table Of ContentA BRAIDED SIMPLICIAL GROUP
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0
0
2 Abstract. By studying braid group actions on Milnor’s construction of the 1-
sphere, we show that the general higher homotopy group of the 3-sphere is the
n
a fixedsetofthepurebraidgroupactiononcertaincombinatoriallydescribedgroup.
J We also give certain representation of higher differentials in the Adams spectral
4 sequence for π∗(S2).
1
]
T
A
1. Introduction
.
h
t In this article, we study the homotopy groups by considering the braid group
a
m actions on simplicial groups. The point of view here is to establish a relation between
the fixed set of braid group actions and the homotopy groups of the 3-sphere. We
[
first recall a combinatorial description of the homotopy groups of the 3-sphere in [11].
1
v Let F(x1,··· ,xn) be the free group generated by the letters x1,··· ,xn. Let
7 w(x ,··· ,x ) = xǫ1 ···xǫt be a word. Given a ,··· ,a ∈ F(x ,··· ,x ), we write
8 w(a1,··· ,an) ∈ F(ix1 ,···it,x ) for aǫ1···aǫt. The1 n-th Wn-group G1 (n) is nthe quotient
0 1 n 1 n i1 it
1 group of F(x ,··· ,x ) modulo the following relations:
1 n
0
0 (R1) the product x1···xn;
0 (R ) the words w(x ,··· ,x ) that satisfy: w(x ,··· ,x ,1,x ,··· ,x ) = 1 for
2 1 n 1 i−1 i+1 n
/
h 1 ≤ i ≤ n.
t
a Relations R consist of all of words that will be trivial if one of the generators is
m 2
replaced by the identity element 1. The smallest normal subgroup of F(x ,··· ,x )
1 n
:
v which contains relations R and R was determined as a subgroup of F(x ,··· ,x )
1 2 1 n
i
X generated by certain systematic and uniform iterated commutators [11].
r Theorem 1.1. [11] For n ≥ 3, the homotopy group πn(S3) is isomorphic to the
a
center of G(n).
A natural question arisen from Theorem 1.1 is how to give a group theoretical
approach to the homotopy groups, that is how to understand the center of the group
G(n). There is a canonical braid group action on G(n) which is induced by the
Research is supported in part by the Academic Research Fund of the National University of
Singapore RP3992646.
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canonical braid group action on the free group F(x ,··· ,x ), namely
1 n
x if j = i
i+1
σ (x ) = x−1 x x if j = i+1
i j i+1 i i+1
xj otherwise
for 1 ≤ i ≤ n − 1. These actions gives a canonical homomorphism from the braid
group B into the automorphism group of G(n). Since Quillen’s plus construction of
n
the classifying space for the stable braid group is (up to homotopy type) the double
loop space of the 3-sphere [2], these braid group actions do not seem occasional
event. Fred Cohen therefore conjectured that the center of G(n) is the fixed set of
the braid group action on G(n). We answer Cohen’s question as follows. Let K
n
be the pure braid group, that is, K is the normal divisor of B generated by σ2
n n 1
(See [6]). Equivalently K is the kernel of the canonical homomorphism from B to
n n
the symmetric group Σ . In geometry, the group K is the fundamental group of the
n n
configuration space F(R2,n), where
F(M,n) = {(x ,··· ,x ) ∈ Mn|x 6= x for i 6= j}
1 n i j
for any manifold M (See [2]). Let Z(G(n)) be the center of G(n).
Theorem 1.2. For n ≥ 4, then
1) the center of G(n) is the fixed set of the pure braid group action on G(n) and
so is π (S3);
n
2) the fixed set of the braid group action on G(n) is the subgroup
{x ∈ Z(G(n))|2x = 0}.
WeshouldpointoutthatthedeterminationofthefixedsetofK -actiononG(n)by
n
(combinatorial) group theoretic means seems beyond the reach of current techniques.
On the other hand, braid group actions have largely 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 certain role for braid group actions. In the range in which π (S3) is
∗
known (∗ ≤ 55, see [5, 9]), by homotopy theoretic means, we gain insight into these
difficult group theoretic questions.
The article is organized as follows. In section 2, we study the braid group action
on the Milnor’s construction on the simplicial 1-sphere. A relation between the sim-
plicial structure and the braid group action is given in Proposition 2.1. This relation
is based on the direct calculation. Roughly the braid group action interchanges the
A BRAIDED SIMPLICIAL GROUP 3
faces together with conjugates. It is possible to have a more general theory to study
group actions on simplicial groups, particularly simplicial group models for iterated
loop spaces. But we only intend to investigate the most important example F(S1)
in this article. After establishing the systematic relations between the braid group
actions and the simplicial structure, braided simplicial groups are introduced in this
section. Then we show that loop simplicial group and the Moore-Postnikov system
of a braided simplicial group are braided. Theorem 2.9 and Proposition 2.10 give a
relation between the fixed set of the braid group action and the homotopy groups for
a general braided simplicial group. Theorem 1.2 follows from Lemma 2.8 and Theo-
rem 2.13. In section 3, we study a braided representation of the Milnor’s construction
of the simplicial 1-sphere into a simplicial algebra. Theorem 3.8 gives certain repre-
sentation of higher differentials in the Adams spectral sequence for π (S2).
∗
The author would like to thanks Professors Fred Cohen and Jon Berrick for their
helpful suggestions and kind encouragements.
2. Braid Group Actions on F(S1)
2.1. Braided simplicial groups. Let F(S1) be Milnor’s construction of the simpli-
cial 1-sphere S1. Then F(S1) = F(y ,··· ,y ) is a free group generated by letters
n+1 0 n
y ,··· ,y with the following simplicial structure
0 n
y for j ≤ k,
k−1
d y = 1 for j = k +1,
j k
y for j > k +1,
k
and
y for j ≤ k,
k+1
s y = y y for j = k +1,
j k k k+1
y for j > k +1,
k
for 0 ≤ j ≤ n+1, where y = (y ···y )−1 in F(S1) (See [11, Lemma 4.1]). Let
−1 0 n−1 n
the braid group B act on F(S1) = F(y ,··· ,y ) in the usual way, that is,
n+1 n+1 0 n
y if j = k
k+1
σ (y ) = y−1 y y if j = k +1
k j k+1 k k+1
y otherwise
j
for 0 ≤ k ≤ n−1. Let σ be an automorphism of F(y ,··· ,y ) defined by
−1 0 n
σ (y ) = y−1y y = y−1y−1···y−1 σ(y ) = y for j > 0.
−1 0 0 −1 0 0 n 1 j j
The subgroup of the automorphism group of F(y ,··· ,y ) generated by σ for −1 ≤
0 n j
j ≤ n−1 is the braid group B . By direct calculation, we have
n+2
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Proposition 2.1. The following identities hold for the braid groups action on F(S1):
σ d j ≤ k
k−1 j
d j = k +1
k+2
(1) d σ =
j k
d j = k +2
k+1
σ d j > k +2
k j
σ s j ≤ k
k+1 j
σ ◦σ ◦s j = k +1
k+1 k k+2
(2) s σ =
j k
σ ◦σ ◦s j = k +2
k k+1 k+1
σ s j > k +2
k j
By this proposition, we give the following definition.
Definition 2.2. A simplicial groupG is called braided if there is a braid group B -
n+1
action on G for each n such that the identities 1 and 2 are satisfied, where B is
n n+1
considered as the braid group generated by σ ,σ ,··· ,σ .
−1 0 n−2
Let G be a simplicial group and let NG be the Moore chain complex of G, that is,
NG = {x ∈ G |d x = 1 for j > 0}
n n j
. Let Z(G) and B(G) be the sets of cycles and boundaries of G, respectively, that is,
Z (G) = {x ∈ G |d x = 1 for all j},
n n j
B (G) = {d x|x ∈ N (G)}.
n 0 n+1
By Moore’s classical theorem [7], π (G) = Z (G)/B (G). Let G be a braided sim-
n n n
plicial group. A subgroup H of G is called a braided subgroup if σ(H) ⊆ H for any
σ ∈ B , that is, H is invariant under B -action.
n+1 n+1
Proposition 2.3. Let G be a braided simplicial group. Then the subgroups Z(G)
n
and B(G) of G are braided.
n n
A BRAIDED SIMPLICIAL GROUP 5
Proof. By Proposition 2.1, Z(G) is a braided subgroup. Now let x = d y ∈ B(G) ,
n 0 n
where y ∈ NG . By Proposition 2.1, we have
n+1
σ x = σ d y = d σ y
k k 0 0 k+1
for each k ≥ −1 and
d σ y = 1
j k+1
for each j > 0 and k ≥ −1. Thus σ x ∈ B(G) for each k ≥ −1 and so B(G) is a
k n n
braided subgroup, which is the assertion.
Note: N (G) is invariant under the subgroup of B generated by σ with k ≥ 0.
n n+1 k
But it is not invariant under σ .
−1
Since there is a relation
d σ = d , d σ = d , and d σ = σ d for j > 1,
0 −1 1 1 −1 0 j −1 −1 j
we have
Proposition 2.4. Let G be a braided simplicial group. Then
Z(G) = NG ∩σ (NG ).
n n −1 n
for each n.
Now we study the braided group actions on the Postnikov systems of a braided
simplicial group G. Let I = (i ,··· ,i ) be a sequence of non-negative integers and
1 k
let d denote the composite of face homomorphisms
I
d = d ···d .
I i1 ik
Given a simplicial group G, the simplicial sub groups R G and R¯ (G) are defined as
n n
follows:
R G = {x ∈ G |d (x) = 1 forany I = (i ,··· ,i )},
n q q I 1 q−n
R¯ G = {x ∈ G |d (x) ∈ B(G) forany I = (i ,··· ,i )}.
n q q I n 1 q−n
Let P G = G/R G and P¯ G = G/R¯ G. Then {P G} is the Postnikov system of G
n n n n n
(See [4, 7]). The quotient homomorphism P G → P¯ G is a homotopy equivalence
n n
(See [11]). The tower
··· → P G → P¯ G → P G → ···
n n n−1
is called a modified Postnikov system of G. One of the important properties of the
modified Postnikov system is that the short exact sequence of simplicial groups
K(π (G),n) → P¯ G → P G
n n n−1
is a central extension [11, Theorem 2.12].
By Propositions 2.1 and 2.3, we have
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Theorem 2.5. Let G be a braided simplicial group. Then, for each n, the simplicial
quotient groups P G and P¯ G are braided. Thus the modified Postnikov system of G
n n
are braided. In particular, there is a braided central extension
¯
K(π (G),n) → P (G) → P (G).
n n n−1
Let G be a braided simplicial group. Then R G is a braided simplicial subgroup of
0
G by the theorem above. Recall that the loop simplicial group ΩG of G is defined by
ΩG = Ker(d )∩R G withd (ΩG) = d (G)ands (ΩG) = s (G)(See[4]). By
n 0 0 n+1 j j+1 j j+1
Proposition2.1, Ker(d )∩R G isinvariantunder theactionofσ for0 ≤ j ≤ n−1.
0 0 n+1 j
Let B′ be the subgroup of B generated by σ ,··· ,σ . Then B′ ∼= B
n+1 n+2 0 n−1 n+1 n+1
under the canonical isomorphism which sends σ to σ for 0 ≤ j ≤ n−1. Thus we
j j−1
obtained the following theorem.
Theorem 2.6. Let G be a braided simplicial group. Then the loop simplicial group
ΩG is braided. Thus any iterated loop simplicial groups of G are braided.
Corollary 2.7. The loops and the modified Postnikov system of F(S1) are braided.
2.2. Fixed Sets of Braided Actions. LetGbeasimplicial groupandletx,y ∈ G
n
be two elements. We call that x is homotopic to y, which is denoted by x ≃ y, if
xy−1 ∈ B(G) .
n
Lemma 2.8. Let G be a braided simplicial group and let x ∈ Z(G) be a cycle with
n
n ≥ 1. Then
σ (x) ≃ x−1
k
for each k ≥ −1.
Proof. For each k ≥ −1, consider the element σ s x. By the identities 1 and 2,
k+1 k+1
we have
σ d s x = 1 for j < k +1
k j k+1
σ x for j = k +1
k
d σ s x = d s x = 1 for j = k +2
j k+1 k+1 k+3 k+1
dk+2sk+1x = x for j = k +3
σk+1djsk+1x = 1 for j > k +3.
Thus σ x ≃ x−1, which is the assertion.
k
A BRAIDED SIMPLICIAL GROUP 7
Let G be a braided simplicial group and let S be a subset of G . Define
n
B (S) = {σ ∈ B |σx ≃ x for all x ∈ S}.
n+1 n+1
Since B(G) is invariant under the braided group action, B (S) is a subgroup of
n+1
B . Let B˜ be the kernel of the composite
n+1 n+1
B - Σ sig-n Z/2,
n+1 n+1
that is, B˜ is the pre-image of the alternating group A .
n+1 n+1
Theorem 2.9. Let G be a braided simplicial group. Then
1) B (Z(G) ) = B˜ or B ;
n+1 n n+1 n+1
2) B (Z(G) ) = B if and only if 2·π (G) = 0.
n+1 n n+1 n
Proof. Let σ ∈ B (Z(G) ) and let x ∈ Z(G) . By Lemma 2.8, we have
n+1 n n
(σ−1σσ )(x) ≃ (σ σ)(x−1) ≃ σ (x−1) ≃ x
k k k k
for each k ≥ −1. Thus B (Z(G) ) is a normal subgroup of B . By Lemma 2.8,
n+1 n n+1
we have
σ σ ∈ B (Z(G) )
s t n+1 n
for any s,t ≥ −1. It follows that B (Z(G) ) = B˜ or B , which is the
n+1 n n+1 n+1
assertion 1.
If B (Z(G) ) = B , then by Lemma 2.8
n+1 n n+1
x ≃ x−1
for any x ∈ Z(G) . Thus 2·π (G) = 0. Conversely, if 2·π (G) = 0, then
n n n
σ (x) ≃ x
k
for any x ∈ Z(G) and k ≥ −1. Thus B (Z(G) ) = B . This shows assertion 2.
n n+1 n n+1
Let G be a braided simplicial group and let H be a subgroup of B . Define
n+1
G (H) = {x ∈ G |σ(x) ≃ x for all σ ∈ H},
n n
that is, G (H) is the “homotopy” fixed set of H. Then G (H) is a subgroup of G .
n n n
By Theorem 2.9, we have that
G (B˜ ) ⊇ Z(G) .
n n+1 n
Proposition 2.10. Let G be a braided simplicial group and let x ∈ G (B˜ ) with
n n+1
n ≥ 2. Then
1) d (x) = d (x) for each j;
j j+2
2) σ d (x) = d (x) for each j,k;
k j j+1
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3) d (x) is a fixed point of B˜ for each j.
j n
In particular, if d x = 1 for some j, then x ∈ Z(G) .
j n
Proof. Since x ∈ G (B˜ ), we have
n n+1
σ σ (x) ≃ x
s t
for any s,t ≥ −1. It follows that
σ x ≃ σ x ≃ σ x ≃ ··· ≃ σ x.
−1 0 1 n−2
Now for each −1 ≤ k ≤ n−3, we have
d x = d σ x = d σ x = d x
k+1 k+2 k k+2 k+1 k+3
and so assertion 1 follows.
Now for each 0 ≤ s ≤ n−2, we have
d (x) = d σ (x) = d σ (x) = σ d (x).
s+1 s s−1 s s s−1 s
Assertion 2 follows.
For any s,t ≥ −1, we have
σ (σ d (x)) = σ (d (σ x)) = d ((σ σ )x) = d x.
s t 0 s 0 t+1 0 t+1 s+1 0
Thus
σ σ (d (x)) = d (x)
t s 0 0
˜
for any s,t ≥ −1 and so d (x) is a fixed point of B . Since
0 n
σ σ (d (x)) = σ ((σ σ )(d (x))) = σ (d (x) = d (x),
t s 1 s −1 t 0 s 0 1
d (x) is a fixed point of B˜ and hence assertion 3.
1 n+1
Let B be the subgroup of B generated by σ with j ≥ 0. By inspecting the
n n+1 j
proof, we have
Proposition 2.11. Let G be a braided simplicial group and let x ∈ G (B˜ ) with
n n
n ≥ 2. Then
1) d (x) = d (x) for each j ≥ 1;
j j+2
2) σ d (x) = d (x) for each j,k ≥ 1;
k j j+1
3) d (x) is a fixed point of B˜ for each j;
j n−1
4) d (x) is a fixed point of B˜ .
0 n
In particular, if d x = 1 and d x = 1 for some j ≥ 1, then x ∈ Z(G) .
0 j n
The following lemma is well-known. We give an elementary proof. Let S be a
subset of a group G. We write hSi for the subgroup of G generated by S.
A BRAIDED SIMPLICIAL GROUP 9
Lemma 2.12. Let w ∈ F(y ,··· ,y ) with n ≥ 0. Suppose that there is a positive
0 n
integer k such that σk(w) = w for 0 ≤ j ≤ n − 1. Then w lies in the subgroup
j
generated by y y ···y . In addition, if σk w = w and n ≥ 1, then w = 1.
0 1 n −1
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 = y−1 and let
0 1
x = y y . Then F(y ,y ) = F(x ,x ). Since σ (y ) = y and σ (y ) = y−1y y , we
1 0 1 0 1 0 1 0 0 1 0 1 1 0 1
have σ = χ and so
0 x1
σk = χt = χ .
0 x1 xt1
We can write w as a reduced word in F(y ,y ) = F(x )∗F(x ). Then
0 1 0 1
w = xn1xl1···xnsxls,
0 1 0 1
where n 6= 0 for 2 ≤ j ≤ s and l 6= 0 for 1 ≤ j ≤ s − 1. Suppose that w 6∈ hx i.
j j 1
Therearetwocases: n 6= 0orn = 0. Ifn 6= 0,thenxtw 6= wxt. Thiscontradictsto
1 1 1 1 1
that χ (w) = w. Otherwise, n = 0 and s > 1. Then w = xl1xn2w′ and xt+l1xn2w′ 6=
xt1 1 1 0 1 0
xl1xl2w′xt. This contradicts to that χ (w) = w. Hence w ∈ hx i = hy y i.
1 0 1 xt1 1 0 1
Now suppose that the assertion holds for n−1 with n > 1. Since
F(y ,··· ,y ) = F(y ,··· ,y )∗F(y )
0 n 0 n−1 n
is a free product, we can write w as a word
w = yl0w yl1···w ylt,
n 1 n t n
where w 6= 1 ∈ F(y ,··· ,y and l 6= 0 for 1 ≤ j ≤ t−1. Because σ (y ) = y
j 0 n−1) j j n n
for j < n−1, we have
σk(w ) = w
j i i
for 1 ≤ i ≤ t and 0 ≤ j ≤ n−2. Let x = y y ···y . By induction, we have
0 1 n−1
w ∈ hxi
i
for 1 ≤ i ≤ t and so
w ∈ hx,y i.
n
Let q: F(y ,y ,··· ,y ) → F(y ,y ) be the projection defined by q(y ) = 1 for
0 1 n n−1 n j
j < n−1 and q(y ) = y for j ≥ n−1. Then
j j
q ◦σ = σ ◦q.
n−1 n−1
Since σ w = w, we have σ (q(w)) = q(w) and so
n−1 n−1
q(w) ∈ hy ·y i.
n−1 n
Because the restriction
q| : hx,y i → F(y ,y )
hx,yni n n−1 n
10 JIE WU
is an isomorphism, we have
w ∈ hx·y i = hy y ···y i
n 0 1 n
and hence the result.
Let K be the pure braided group, that is, K is that kernel of the canonical
n n
epimorphism B → Σ . Let B be the subgroup of B generated by σ for j ≥ 0.
n n n+1 n+2 j
Recall that
π (F(S1)) = Z(F(S1)) /B(F(S1)) .
n+1 n+1 n+1
ConsidertheactionsoftwobraidedgroupsB andB onF(S1) /B(F(S1)) .
n+2 n+1 n+1 n+1
We have
Theorem 2.13. If n ≥ 2, then in F(S1) /B(F(S1)) ,
n+1 n+1
1) the fixed set of the pure braided group K -action is
n+1
Z⊕π (F(S1));
n+1
2) the fixed set of K -action is
n+2
π (F(S1)).
n+1
Proof. We show that the homotopy fixed set of K on F(S1) is generated by
n+1 n+1
y and Z(F(S1)) . Assertions 1 and 2 will follow from this statement. Let w be
−1 n+1
a homotopy fixed point of K -action on F(S1) . Since
n+1 n+1
σ2d = d σ2
k 0 0 k+1
for k ≥ −1, we have
σ2d (w) = d (w)
k 0 0
for each k ≥ −1. By Lemma 2.12, we have
d (w) = 1.
0
Now for each 1 ≤ j ≤ n+1, we have
d σ2 if j ≤ k +1;
j k+1
σ2d = d ◦σ−1 ◦σ2 ◦σ if j = k +2;
k j j j−1 j−2 j−1
d σ2 if j > k +2.
j k
By Lemma 2.12, there exists integers k ,k ,··· ,k such that
1 2 n+1
d (w) = ykj
j −1
for 1 ≤ j ≤ n+1. Since d y = y for k > 0, we have
k −1 −1
ykj = d (ykj ) = d d w = d d w = d (ykj+1) = ykj+1
−1 j −1 j j j j+1 j −1 −1
for 1 ≤ j ≤ n and so
k = k = ··· = k .
1 2 n+1
