A BRAIDED SIMPLICIAL GROUP
JIE WU
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
fixed set of the pure braid group action on certain combinatorially descr*
*ibed group.
We also give certain representation of higher differentials in the Adams *
*spectral
sequence for ss*(S2).
1. Introduction
In this article, we study the homotopy groups by considering the braid group
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].
Let F (x1; . .;.xn) be the free group generated by the letters x1; . .;.xn. *
*Let
w(x1; . .;.xn) = xffl1i1.x.f.fltitbe a word. Given a1; . .;.an 2 F (x1; . .;.x*
*n), we write
w(a1; . .;.an) 2 F (x1; . .;.xn) for affl1i1.a.f.fltit. The n-th W -group G(n) *
*is the quotient
group of F (x1; . .;.xn) modulo the following relations:
(R1) the product x1. .x.n;
(R2) the words w(x1; . .;.xn) that satisfy: w(x1; . .;.xi-1; 1; xi+1; . .;.xn)*
* = 1 for
1 i n.
Relations R2 consist of all of words that will be trivial if one of the gener*
*ators is
replaced by the identity element 1. The smallest normal subgroup of F (x1; . .;*
*.xn)
which contains relations R1 and R2 was determined as a subgroup of F (x1; . .;.*
*xn)
generated by certain systematic and uniform iterated commutators [11].
Theorem 1.1. [11] For n 3, the homotopy group ssn(S3) is isomorphic to the
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 gr*
*oup
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 U*
*niversity of
Singapore RP3992646.
1
2 JIE WU
canonical braid group action on the free group F (x1; . .;.xn), namely
8
>> xi+1 if j = i
>>
><
oei(xj) = x-1i+1xixi+1ifj = i + 1
>>
>>
>:
xj otherwise
for 1 i n - 1. These actions gives a canonical homomorphism from the braid
group Bn 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 d*
*ouble
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 se*
*t of
the braid group action on G(n). We answer Cohen's question as follows. Let Kn
be the pure braid group, that is, Kn is the normal divisor of Bn generated by o*
*e21
(See [6]). Equivalently Kn is the kernel of the canonical homomorphism from Bn *
*to
the symmetric group n. In geometry, the group Kn is the fundamental group of the
configuration space F (R2; n), where
F (M; n) = {(x1; . .;.xn) 2 Mn|xi6= xj fori 6= 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 ssn(S3);
2) the fixed set of the braid group action on G(n) is the subgroup
{x 2 Z(G(n))|2x = 0}:
We should point out that the determination of the fixed set of Kn-action on G*
*(n) by
(combinatorial) group theoretic means seems beyond the reach of current techniq*
*ues.
On the other hand, braid group actions have largely studied in several areas su*
*ch 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 homot*
*opy
groups play certain role for braid group actions. In the range in which ss*(S3*
*) is
known (* 55, see [5, 9]), by homotopy theoretic means, we gain insight into th*
*ese
difficult group theoretic questions.
The article is organized as follows. In section 2, we study the braid group a*
*ction
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 it*
*erated
loop spaces. But we only intend to investigate the most important example F (S*
*1)
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 giv*
*e a
relation between the fixed set of the braid group action and the homotopy group*
*s 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 const*
*ruction
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 ss*(S2).
The author would like to thanks Professors Fred Cohen and Jon Berrick for the*
*ir
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 simp*
*li-
cial 1-sphere S1. Then F (S1)n+1 = F (y0; . .;.yn) is a free group generated by*
* letters
y0; . .;.yn with the following simplicial structure
8
< yk-1 for j k;
djyk = 1 for j = k + 1;
: y
k for j > k + 1;
and 8
< yk+1 for j k;
sjyk = ykyk+1 for j = k + 1;
: y
k for j > k + 1,
for 0 j n + 1, where y-1 = (y0. .y.n-1)-1 in F (S1)n (See [11, Lemma 4.1]). L*
*et
the braid group Bn+1 act on F (S1)n+1 = F (y0; . .;.yn) in the usual way, that *
*is,
8
< yk+1 if j = k
oek(yj) = y-1k+1ykyk+1 if j = k + 1
: y
j otherwise
for 0 k n - 1. Let oe-1 be an automorphism of F (y0; . .;.yn) defined by
oe-1(y0) = y-10y-1y0 = y-10y-1n. .y.-11oe(yj) = yj for j > 0:
The subgroup of the automorphism group of F (y0; . .;.yn) generated by oej for *
*-1
j n - 1 is the braid group Bn+2. By direct calculation, we have
4 JIE WU
Proposition 2.1. The following identities hold for the braid groups action on F*
* (S1):
8
>>oek-1dj j k
>>
>>
>>
>>
>< dk+2 j = k + 1
(1) djoek =
>>
>> d j = k + 2
>> k+1
>>
>>
>:
oekdj j > k + 2
8
>> oek+1sj j k
>>
>>
>>
>>
>>
>>oe O oe O s j = k + 2
>> k k+1 k+1
>>
>>
>:
oeksj j > k + 2
By this proposition, we give the following definition.
Definition 2.2. A simplicial group G is called braided if there is a braid grou*
*p Bn+1-
action on Gn for each n such that the identities 1 and 2 are satisfied, where B*
*n+1 is
considered as the braid group generated by oe-1; oe0; . .;.oen-2.
Let G be a simplicial group and let NG be the Moore chain complex of G, that *
*is,
NGn = {x 2 Gn|djx = 1 for j > 0}
. Let Z(G) and B(G) be the sets of cycles and boundaries of G, respectively, th*
*at is,
Zn(G) = {x 2 Gn|djx = 1 for all j};
Bn(G) = {d0x|x 2 Nn+1(G)}:
By Moore's classical theorem [7], ssn(G) = Zn(G)=Bn(G). Let G be a braided sim-
plicial group. A subgroup H of G is called a braided subgroup if oe(H) H for a*
*ny
oe 2 Bn+1, that is, H is invariant under Bn+1-action.
Proposition 2.3. Let G be a braided simplicial group. Then the subgroups Z(G)n
and B(G)n of Gn are braided.
A BRAIDED SIMPLICIAL GROUP 5
Proof.By Proposition 2.1, Z(G)n is a braided subgroup. Now let x = d0y 2 B(G)n,
where y 2 NGn+1. By Proposition 2.1, we have
oekx = oekd0y = d0oek+1y
for each k -1 and
djoek+1y = 1
for each j > 0 and k -1. Thus oekx 2 B(G)n for each k -1 and so B(G)n is a *
* __
braided subgroup, which is the assertion. *
*|__|
Note: Nn(G) is invariant under the subgroup of Bn+1 generated by oek with k 0.
But it is not invariant under oe-1.
Since there is a relation
d0oe-1 = d1; d1oe-1 = d0; and djoe-1 = oe-1dj for j > 1;
we have
Proposition 2.4. Let G be a braided simplicial group. Then
Z(G)n = NGn \ oe-1(NGn):
for each n.
Now we study the braided group actions on the Postnikov systems of a braided
simplicial group G. Let I = (i1; . .;.ik) be a sequence of non-negative integer*
*s and
let dI denote the composite of face homomorphisms
dI = di1. .d.ik:
Given a simplicial group G, the simplicial sub groups RnG and Rn(G) are defined*
* as
follows:
RnGq = {x 2 Gq|dI(x) = 1 forany I = (i1; . .;.iq-n)};
RnGq = {x 2 Gq|dI(x) 2 B(G)n forany I = (i1; . .;.iq-n)}:
Let PnG = G=RnG and PnG = G=Rn G. Then {PnG} is the Postnikov system of G
(See [4, 7]). The quotient homomorphism PnG ! PnG is a homotopy equivalence
(See [11]). The tower
. .!.PnG ! PnG ! Pn-1G ! . . .
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(ssn(G); n) ! PnG ! Pn-1G
is a central extension [11, Theorem 2.12].
By Propositions 2.1 and 2.3, we have
6 JIE WU
Theorem 2.5. Let G be a braided simplicial group. Then, for each n, the simpl*
*icial
quotient groups PnG and PnG are braided. Thus the modified Postnikov system of G
are braided. In particular, there is a braided central extension
K(ssn(G); n) ! Pn(G) ! Pn-1(G):
Let G be a braided simplicial group. Then R0G is a braided simplicial subgrou*
*p of
G by the theorem above. Recall that the loop simplicial group G of G is defined*
* by
Gn = Ker(d0)\R0Gn+1 with dj(G) = dj+1(G) and sj(G) = sj+1(G) (See [4]). By
Proposition 2.1, Ker(d0)\R0Gn+1 is invariant under the action of oej for 0 j *
*n-1.
Let B0n+1be the subgroup of Bn+2 generated by oe0; . .;.oen-1. Then B0n+1~=Bn+1
under the canonical isomorphism which sends oej to oej-1 for 0 j n - 1. Thus *
*we
obtained the following theorem.
Theorem 2.6. Let G be a braided simplicial group. Then the loop simplicial gr*
*oup
G 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 braide*
*d.
2.2. Fixed Sets of Braided Actions. Let G be a simplicial group and let x; y 2 *
*Gn
be two elements. We call that x is homotopic to y, which is denoted by x ' y, *
*if
xy-1 2 B(G)n.
Lemma 2.8. Let G be a braided simplicial group and let x 2 Z(G)n be a cycle w*
*ith
n 1. Then
oek(x) ' x-1
for each k -1.
Proof.For each k -1, consider the element oek+1sk+1x. By the identities 1 and *
*2,
we have 8
>> oekdjsk+1x = 1 for j < k + 1
>>
>>
>>
>>
>> oekx for j = k + 1
>>
><
djoek+1sk+1x = dk+3sk+1x = 1 for j = k + 2
>>
>>
>>
>>
>> dk+2sk+1x = x for j = k + 3
>>
>>
>:
oek+1djsk+1x = 1 for j > k + 3:
*
* __
Thus oekx ' x-1, which is the assertion. *
*|__|
A BRAIDED SIMPLICIAL GROUP 7
Let G be a braided simplicial group and let S be a subset of Gn. Define
Bn+1(S) = {oe 2 Bn+1|oex ' x for all x 2 S}:
Since B(G) is invariant under the braided group action, Bn+1(S) is a subgroup of
Bn+1. Let "Bn+1be the kernel of the composite
sign
Bn+1 ____-n+1 ____-Z=2;
that is, "Bn+1is the pre-image of the alternating group An+1.
Theorem 2.9. Let G be a braided simplicial group. Then
1) Bn+1(Z(G)n) = "Bn+1or Bn+1;
2) Bn+1(Z(G)n) = Bn+1 if and only if 2 . ssn(G) = 0.
Proof.Let oe 2 Bn+1(Z(G)n) and let x 2 Z(G)n. By Lemma 2.8, we have
(oe-1koeoek)(x) ' (oekoe)(x-1) ' oek(x-1) ' x
for each k -1. Thus Bn+1(Z(G)n) is a normal subgroup of Bn+1. By Lemma 2.8,
we have
oesoet2 Bn+1(Z(G)n)
for any s; t -1. It follows that Bn+1(Z(G)n) = B"n+1 or Bn+1, which is the
assertion 1.
If Bn+1(Z(G)n) = Bn+1, then by Lemma 2.8
x ' x-1
for any x 2 Z(G)n. Thus 2 . ssn(G) = 0. Conversely, if 2 . ssn(G) = 0, then
oek(x) ' x
for any x 2 Z(G)n and k -1. Thus Bn+1(Z(G)n) = Bn+1. This shows assertion 2._
|__|
Let G be a braided simplicial group and let H be a subgroup of Bn+1. Define
Gn(H) = {x 2 Gn|oe(x) ' x for all oe 2 H};
that is, Gn(H) is the "homotopy" fixed set of H. Then Gn(H) is a subgroup of Gn.
By Theorem 2.9, we have that
Gn(B"n+1) Z(G)n:
Proposition 2.10. Let G be a braided simplicial group and let x 2 Gn(B"n+1) wi*
*th
n 2. Then
1) dj(x) = dj+2(x) for each j;
2) oekdj(x) = dj+1(x) for each j; k;
8 JIE WU
3) dj(x) is a fixed point of "Bnfor each j.
In particular, if djx = 1 for some j, then x 2 Z(G)n.
Proof.Since x 2 Gn(B"n+1), we have
oesoet(x) ' x
for any s; t -1. It follows that
oe-1x ' oe0x ' oe1x ' . .'.oen-2x:
Now for each -1 k n - 3, we have
dk+1x = dk+2oekx = dk+2oek+1x = dk+3x
and so assertion 1 follows.
Now for each 0 s n - 2, we have
ds+1(x) = dsoes-1(x) = dsoes(x) = oes-1ds(x):
Assertion 2 follows.
For any s; t -1, we have
oes(oetd0(x)) = oes(d0(oet+1x)) = d0((oet+1oes+1)x) = d0x:
Thus
oetoes(d0(x)) = d0(x)
for any s; t -1 and so d0(x) is a fixed point of "Bn. Since
oetoes(d1(x)) = oes((oe-1oet)(d0(x))) = oes(d0(x) = d1(x);
*
* __
d1(x) is a fixed point of "Bn+1and hence assertion 3. *
* |__|
Let Bn be the subgroup of Bn+1 generated by oej with j 0. By inspecting the
proof, we have
Proposition 2.11. Let G be a braided simplicial group and let x 2 Gn(B"n) with
n 2. Then
1) dj(x) = dj+2(x) for each j 1;
2) oekdj(x) = dj+1(x) for each j; k 1;
3) dj(x) is a fixed point of "Bn-1for each j;
4) d0(x) is a fixed point of "Bn.
In particular, if d0x = 1 and djx = 1 for some j 1, then x 2 Z(G)n.
The following lemma is well-known. We give an elementary proof. Let S be a
subset of a group G. We write ~~ for the subgroup of G generated by S.
A BRAIDED SIMPLICIAL GROUP 9
Lemma 2.12. Let w 2 F (y0; . .;.yn) with n 0. Suppose that there is a posit*
*ive
integer k such that oekj(w) = w for 0 j n - 1. Then w lies in the subgroup
generated by y0y1. .y.n. In addition, if oek-1w = w and n 1, then w = 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 x0 = y-11and let
x1 = y0y1. Then F (y0; y1) = F (x0; x1). Since oe0(y0) = y1 and oe0(y1) = y-11y*
*0y1, we
have oe0 = Ox1 and so
oek0= Otx1= Oxt1:
We can write w as a reduced word in F (y0; y1) = F (x0) * F (x1). Then
w = xn10xl11. .x.ns0xls1;
where nj 6= 0 for 2 j s and lj 6= 0 for 1 j s - 1. Suppose that w 62 .
There are two cases: n1 6= 0 or n1 = 0. If n1 6= 0, then xt1w 6= wxt1. This con*
*tradicts to
that Oxt1(w) = w. Otherwise, n1 = 0 and s > 1. Then w = xl11xn20w0and xt+l11xn2*
*0w06=
xl11xl20w0xt1. This contradicts to that Oxt1(w) = w. Hence w 2 = .
Now suppose that the assertion holds for n - 1 with n > 1. Since
F (y0; . .;.yn) = F (y0; . .;.yn-1) * F (yn)
is a free product, we can write w as a word
w = yl0nw1yl1n. .w.tyltn;
where wj 6= 1 2 F (y0; . .;.yn-1) and lj 6= 0 for 1 j t - 1. Because oej(yn) *
*= yn
for j < n - 1, we have
oekj(wi) = wi
for 1 i t and 0 j n - 2. Let x = y0y1. .y.n-1. By induction, we have
wi2
for 1 i t and so
w 2 :
Let q :F (y0; y1; . .;.yn) ! F (yn-1; yn) be the projection defined by q(yj) = *
*1 for
j < n - 1 and q(yj) = yj for j n - 1. Then
q O oen-1 = oen-1 O q:
Since oen-1w = w, we have oen-1(q(w)) = q(w) and so
q(w) 2 :
Because the restriction
q|: ! F (yn-1; yn)
10 JIE WU
is an isomorphism, we have
w 2 =
__
and hence the result. |_*
*_|
Let Kn be the pure braided group, that is, Kn is that kernel of the canonical
epimorphism Bn ! n. Let Bn+1 be the subgroup of Bn+2 generated by oej for j 0.
Recall that
ssn+1(F (S1)) = Z(F (S1))n+1=B(F (S1))n+1:
Consider the actions of two braided groups Bn+2 and Bn+1 on F (S1)n+1=B(F (S1))*
*n+1.
We have
Theorem 2.13. If n 2, then in F (S1)n+1=B(F (S1))n+1,
1) the fixed set of the pure braided group Kn+1-action is
Z ssn+1(F (S1));
2) the fixed set of Kn+2-action is
ssn+1(F (S1)):
Proof.We show that the homotopy fixed set of Kn+1 on F (S1)n+1 is generated by
y-1 and Z(F (S1))n+1. Assertions 1 and 2 will follow from this statement. Let w*
* be
a homotopy fixed point of Kn+1-action on F (S1)n+1. Since
oe2kd0 = d0oe2k+1
for k -1, we have
oe2kd0(w) = d0(w)
for each k -1. By Lemma 2.12, we have
d0(w) = 1:
Now for each 1 j n + 1, we have
8
< djoe2k+1 if j k + 1;
oe2kdj = dj O oe-1j-1O oe2j-2O oej-1ifj = k + 2;
: d 2
joek if j > k + 2:
By Lemma 2.12, there exists integers k1; k2; . .;.kn+1 such that
dj(w) = ykj-1
for 1 j n + 1. Since dky-1 = y-1 for k > 0, we have
ykj-1= dj(ykj-1) = djdjw = djdj+1w = dj(ykj+1-1) = ykj+1-1
for 1 j n and so
k1 = k2 = . .=.kn+1:
A BRAIDED SIMPLICIAL GROUP 11
Let w0= y-k1-1w. Then
dj(w0) = 1
for each 0 j n + 1 and w02 Z(F (S1))n+1. This shows that w lies in the subgro*
*up__
generated by y-1 and cycles Z(F (S1))n+1 and hence the result. *
* |__|
Note: In F (S1)n+1=Bn+1, since any element in the homotopy group is a homotopy
fixed point of B"n+2, the fixed set of B"n+1is Z ssn+1(F (S1)) and the fixed s*
*et of
"Bn+2is ssn+1(F (S1)).
3.Braided Representation of F (S1)
3.1. A Representation of F (S1). In this subsection, the ground ring R is Z or
Z=p. Let X be a pointed set. Let A(X) be the algebra of non-commutative formal
power series with variables in any finite subset of X over R modulo the single *
*relation
that * = 1, where * is the base point of X. Let X be a pointed simplicial set. *
*The
simplicial algebra A(X) is defined by applying the functor A to X. Let X = S1 be
the simplicial circle. By using the methods in [11], we have
1) There is a choice of generators in A(S1)n+1 such that
A(S1)n+1 = A(x0; x1; . .;.xn)
is the associated algebra of the non-commutative formal power series in v*
*ari-
ables x0; x1; . .;.xn over R.
2) The simplicial structure on A(S1) is given by
8
< xk-1 for j k;
djxk = 0 for j = k + 1;
: x
k for j > k + 1;
and 8
< xk+1 for j k;
sjxk = xk + xk+1 + xkxk+1 for j = k + 1;
: x
k for j > k + 1,
for 0 j n + 1, where
x-1 = (1 + xn-1)-1(1 + xn-2)-1 . .(.1 + x0)-1 - 1
in A(S1)n.
Let the Braided group Bn+1 act on A(S1)n+1 as follows.
12 JIE WU
For each 0 k n - 1, oek: A(S1)n+1 ! A(S1)n+1 is an automorphism of
algebras with
8
< xk+1 if j = k
oek(xj) = (1 + xk+1)-1(1 + xk)(1 + xk+1) - 1 if j = k + 1
: x
j otherwise
for 0 k n - 1.
Let oe-1 be an automorphism of A(S1)n+1 defined by
oe-1(x0) = (1 + x0)-1(1 + x-1)(1 + x0) - 1 = (1 + x0)-1(1 + xn)-1 . .(.1 + x1)-*
*1 - 1
and oe(xj) = xj for j > 0. The subgroup of the automorphism group of A(S1)n+1
generated by oej for -1 j n - 1 is the braid group Bn+2. Let
e: F (S1) ! A(S1)
be the canonical representation, that is
e(yj) = 1 + xj
for each j.
Proposition 3.1. The simplicial algebra A(S1) is a braided simplicial algebra, *
*that
is, the braided action satisfies the identities 1 and 2 in Proposition 2.1. Fur*
*thermore,
the representation e: F (S1) ! A(S1) is a braided representation, that is e com*
*mutes
with the braid group action and the simplicial structure.
The proof is straight forward.
Since the representation e: F (S1) ! A(S1) is faithful [6], we have
Proposition 3.2. Let w 2 F (S1)n+1. Then
1) w 2 Z(F (S1))n+1 if and only if e(w) - 1 2 Z(A(S1))n+1;
2) w 2 NF (S1)n+1 if and only if e(w) - 1 2 NA(S1)n+1.
Let w = xn1i1xn2i2.x.n.titbe a monomial in A(x0; . .;.xn). We call w is non-d*
*egenerate
if the set
{i1; . .;.it} = {0; . .;.n};
that is each letter xj with 0 j n appears in w at least once.
Theorem 3.3. Let f be a series in A(S1)n+1. Then f 2 NA(S1)n+1 if and only if
f is a formal series of non-degenerate monomials.
The assertion follows from the following lemma.
A BRAIDED SIMPLICIAL GROUP 13
Lemma 3.4. Let f be a series in A(S1)n+1. Then
i+1"
f 2 Ker(dj)
j=1
if and only if f is a linear summation of monomials in which each xj appears at*
* least
once for 0 j i.
Proof.The proof is given by induction on i. Let i = 0. Since d1 is the projec*
*tion,
Ker(d0) is a two sided ideal generated by x0. Suppose that the assertion holds*
* for
i - 1 with i > 0. Then there is a decomposition
"i
Ker dj = C D;
j=1
where C is the set of series of monomials in which each xj appears at least onc*
*e for
0 j i. Since di+1 is the projection, we find that di+1|C = 0 and di+1|D is a
monomorphism from D to A(S1)n. This shows that
i+1"
Ker dj = C
j=1
__
and hence the result. |_*
*_|
3.2. Formal Steenrod Operations and Higher Differentials. In this subsec-
tion, the ground ring R is Z or Z=p. Let A(x0; . .;.xn)i be the sub R-module of
A(x0; . .;.xn) generated by monomial of degree i. Let f :A(x0; . .;.xn) ! A(x0;*
* . .;.xn-1)
be an R-linear map. We call f is a homogenous map of degree t if
f(A(x0; . .;.xn)i) A(x0; . .;.xn-1)i+t
for each q. Let V be the free R-module generated by x0; . .;.xn.
Lemma 3.5. Let ft:V ! A(x0; . .x.n-1) be a sequence of R-linear maps with t 0
such that
ft(V ) A(x0; . .;.xn-1)t
for each t. Then there exists a unique sequence of homogeneous maps Pft:A(x0; .*
* .;.xn) !
A(x0; . .;.xn-1) such that
1) Pft|V = ft for each t 0;
2) The anti-Cartan formula
X
Pft(xy) = Pfi(y)Pfj(x)
i+j=t
hold for any x; y 2 A(x0; . .;.xn).
14 JIE WU
The proof is straight forward.
Let O: A(x0; . .;.xn) be the convolution, that is, O is the anti-automorphism*
* with
O(xi) = -xi for each i. Let
d0= d0 O O = O O d0:
Then we have d0(xj) = xj-1 for j > 0 and
d0(x0) = (1 + x0)(1 + x1) . .(.1 + xn-1) - 1:
Let X
s-1 = xl1xl2. .x.ls:
0l1 0;
ffii(xj) = 0 for i > 0; j > 0
:
i for j = 0
Let
ffi = (ffi0; ffi1; . .;.ffin-1; 0; . .;.0):
Proposition 3.6. The map d0:A(x0; . .;.xn) ! A(x0; . .;.xn-1) is decomposed as
X1
d0 = Pfifi:
i=0
Proof.By the anti-Cartan formula, the map
X1
Pfifi
i=0
is a well-defined anti-homomorphism of algebras. Since
X1
d0(xj) = Pfifi(xj)
i=0
*
*__
for each 0 j n, the assertion follows. |*
*__|
Let qi:A(x0; . .;.xn) ! A(x0; . .;.xn) be the composite
proj:
A(x0; . .;.xn) ____-A(x0; . .;.xn)i ____-A(x0; . .;.xn)
for 0 i < 1. Let
tF (S1)n+1 = {w 2 F (S1)n+1|qi(w) = 0 for 0 < i < t}
A BRAIDED SIMPLICIAL GROUP 15
for 1 t < 1. It is well-known [6] that {t} is the descending central series if*
* R = Z
and descending p-central series if R = Z=p. Let
M1
L(S1) = t(F (S1))=t+1(F (S1)):
t=1
If R = Z, then L(S1) is the free simplicial Lie algebra over Z generated by S1.*
* If
R = Z=p, then L(S1) is the free restricted simplicial Lie algebra over Z=p gene*
*rated
by S1. The simplicial structure L(S1) is as follows.
1) L(S1)n+1 = L(x0; . .;.xn);
2) dj(xk) = xk for k < j - 1, dj(xj-1) = 0 and dj(xk) = xk-1 for k > j - 1, *
*where
x-1 = -(x0 + x1 + . .+.xn-1) 2 Lu(S1)n or L(S1)n.
3) sjxk = xk for k < j - 1, sjxj-1 = xj-1+ xj and sjxk = xk+1 for k > j - 1.
s u 1 1
A Lie monomial w = [[xi1; xi2; . .].pin L (S )n+1 or L(S )n+1 is called non-deg*
*enerate
if the set
{i1; . .;.it} = {0; 1; . .;.n}:
By Theorem 3.3, we have
Corollary 3.7. The Moore chain complexes NL(S1)n+1 is the submodules of L(S1)n+1
spanned by non-degenerate Lie monomials, respectively.
The spectral sequence, which is denoted by {Er(F (S1)}, induced by the descen*
*ding
p-central (integral descending central) series of F (S1) is called (integral) A*
*dams spec-
tral sequence of F (S1). By Curtis Theorem , this spectral sequence is converge*
*nt to
ss*(F (S1); p) = ss*+1(S2; p) ( or ss*+1(S2) if we use integral descending cent*
*ral series).
A description of higher differentials in the Adams spectral sequence is as foll*
*ows.
Let w 2 F (S1)n+1. Then
X1
e(w) = 1 + qi(w)
i=1
in A(x0; . .;.xn). We simply write (w)i for qi(w). Let
ff 2 E1t;*(F (S1)) = ss*(Lt(S1))
and let zffbe a cycle in the simplicial group L(S1) such that the homotopy clas*
*s of
zffis ff, that is zffis a cycle representative of ff. Since the map
g :t(F (S1)) ! Lt(S1) = t(F (S1))=t+1(F (S1));
is a simplicial epimorphism, there is an element
wff2 N(t(F (S1))) = N(F (S1)) \ F (S1)
such that
g(wff) = zff:
16 JIE WU
The element wffis called a Moore representative of ff.
Theorem 3.8. Let ff 2 E1t;*(F (S1n+1)) and let wffbe a Moore representative*
* in
F (S1)n+1. Let 1 r 1. Then dj(ff) = 0 for j < r if and only if the follow-
ing linear equations holds in A(x0; . .;.xn-1):
Xj
Pfifiwfft+j-i= 0:
i=0
for 0 j < r. Furthermore, if r < 1, then
X r
Pfifiwfft+r2 Lt+r(S1)n A(x0; . .;.xn-1);
i=0
which is a cycle representative of -dr(ff).
Proof.Since dr(ff) = 0 if and only if
d0(wff) 2 t+rF (S1)n;
*
* __
the assertion follows from Proposition 3.6. *
* |__|
Let w be a word in F (S1)n+1. We call w is a basic non-degenerate commutator *
*of
weight s = l(w) if w can be written down as a commutator
[. .[.yi1; yi2]; . .].; yis]
such that the set
{i1; . .;.is} = {0; . .;.n};
that is each generator yj occurs in the commutator w at least once. Let NtF (S1*
*)n+1
be the subgroup of F (S1)n+1 generated by basic nondegenerate commutators w with
r
l(w) t and let N(p)tF (S1)n+1 be the subgroup of F (S1)n+1 generated by wp ; w*
*here
w runs over all basic non-degenerated commutators with l(w) . pr t. Note that
NtF (S1)n+1 N(F (S1))n+1 \ t(F (S1)n+1)
if R = Z and
N(p)tF (S1)n+1 N(F (S1))n+1 \ t(F (S1)n+1)
if R = Z=p.
By Corollary 3.7, we have
Proposition 3.9. Let r be any non-negative integer. If R = Z, there is an isomo*
*r-
phism
NtF (S1)n+1=(NtF (S1)n+1 \ t+rF (S1)n+1)
~=N(F (S1))n+1 \ t(F (S1))n+1=(N(F (S1))n+1 \ t+r(F (S1))n+1):
A BRAIDED SIMPLICIAL GROUP 17
If R = Z=p, then there is an isomorphism
N(p)tF (S1)n+1=(N(p)tF (S1)n+1 \ t+rF (S1)n+1)
~=N(F (S1))n+1 \ t(F (S1))n+1=(N(F (S1))n+1 \ t+r(F (S1))n+1):
Remark 3.10. By Theorem 3.8 and Proposition 3.9, ss*(Lt(S1)) are represented *
*by
those words w in NtF (S1) (or N(p)tF (S1)) with Pf0fiwt= 0 and the higher diffe*
*rentials
in the Adams spectral sequence are related to higher formal Steenrod operations*
* on
NtF (S1).
3.3. E1-terms of The Integral Adams Spectral Sequence. In this subsection,
the ground ring R is a subring of Q. Let L be the free functor from free R-modu*
*les
to Lie algebras. Let X be a pointed simplicial set. Let R (X) = R(X)=R(*) be the
reduced free simplicial R-module generated by X. In particular, R (Sn) = K(R; n*
*).
Let L(X) = L(R (X)) for any pointed simplicial set X. Let V be a free R-module
and let L0be the kernel of the abelianizer
L(V ) ! V:
Then L0 is functor from free R-modules to graded free R-modules. Let Qn(L0(V ))
be the set of indecomposable elements of degree n of L0(V ). Let Sn(V ) be the *
*set of
monomials of degree n in the polynomial algebra S(V ). Note that Q1(L0(V )) = 0.
Lemma 3.11. For each n 2, there is a functorial short exact sequence
mult:
0 ! Qn(L0(V )) ! Sn-1(V ) V ____-Sn(V ) ! 0:
mult
Proof.Let K(V ) be the kernel of Sn-1(V )V ____-Sn(V ) and let OE: Tn(V ) ! Sn(*
*V )
be the composite
proj: mult:
Tn(V ) = Tn-1(V ) V ____-Sn-1(V ) V ____-Sn(V ):
Then OE|L0n(V:)L0n(V ) ! Sn(V ) is zero which gives a functorial map
"OE:L0n(V ) ! K(V ):
It is a routine work to show that the map "OEis onto and "OE|Dn(L0(V )):Dn(L0(V*
* )) !
K(V ) is zero and so O"Efactors through Qn(L0(V )), where D(L0(V )) is the set *
*of
decomposable elements of L0(V ). Thus K(V ) is a functorial quotient of Qn(L0(V*
* )).
By checking Poincare series, the quotient map
Qn(L0(V )) ! K(V )
*
*__
is an isomorphism and hence the result. |*
*__|
Note: There is no functorial cross-section from Q(L0) to L0(See [8]).
18 JIE WU
Proposition 3.12. There is a homotopy equivalence
L(S1) ' R(S1) L(S2):
Proof.Since K(Z; 1) ' S1, we have
Q(L0) ' R(S2):
Let OE: L(L2(S1)) ! L0(S1) be the inclusion. Then OE is a homotopy equivalence *
*by __
checking the spectral sequence induced by Lie filtrations. The assertion follow*
*s. |__|
Let V be a free simplicial R-module and let C be a (pointed) simplicial coalg*
*ebra.
Let f; g :C ! T (V ) be pointed simplicial coalgebra maps. We call f is coalge*
*bra
homotopic to g if there is a pointed homotopy Ft:C ! T (V ) such that F0 = f,
F1 = g and Ft is a coalgebra map for each t. Let
[T (V ); T (V )]coalg
be the set of coalgebra homotopy classes. If f :T (V ) ! T (V ) is a simplicial*
* coalgebra
map, then we have the restriction
f|L(V ):L(V ) ! L(V ):
If f is coalgebra homotopic to g, then
f|L(V )' g|L(V ):
This defines a map
:[T (V ); T (V )]coalg! [L(V ); L(V )]:
Theorem 3.13. Suppose that R = Z(p). Then the homotopy groups ss*(L(S2n)) has
the following exponents.
1) If n = 1; 3, then
p . ss*(L(S2n)) = 0
for * > 2n and any prime p;
2) If p = 2, then
4 . ss*(Lt(S2n)) = 0
for t > 2 and any n.
Proof.Consider the Cohen representation
:H1 ! [T (S2n); T (S2n)]coalg! [L(S2n); L(S2n)];
where H1 is the Cohen group. First we assume that n = 1; 3. Then the Samelson
product
S2n ^ S2n ! F (S2n)
A BRAIDED SIMPLICIAL GROUP 19
is null homotopic. Since ss*(2(F (S2n)) ! ss*(F (S2n)) is a monomorphism, the
Samelson product
S2n ^ S2n ! 2(F (S2n))
is null homotopic and so the composite
S2n ^ S2n ! 2(F (S2n)) ! 2(F (S2n)=3(F (S2n))
is null homotopic. Let JtT (S2n) = Z(p)(Jt(S2n)). Then there is a monomorphism
[JtT (S2n); T (S2n)]coalg! [Z(p)((S2n)xt); T (S2n)]coalg
for any t. Since the Samelson product is trivial, the group
[Z(p)((S2n)xt); T (S2n)]coalg
is abelian. Let id*p:T (V ) ! T (V ) be the p-fold convolution product of the i*
*dentity
and let T (p): T (V ) ! T (V ) be the morphism of Hopf algebras induced by the *
*map
p: V ! V; ; x ! px. Then id*p|JtT(S2n)is coalgebra homotopic to T (p)|JtT(S2n*
*)in
[Z(p)((S2n)xt); T (S2n)]coalg
for each t and so
id*p|JtT(S2n)' T (p)JtT(S2n)
for each t. This id*pis coalgebra homotopic to T (p). Let x 2 Lt(S2n). Then
id*p(x) = px T (p)(x) = ptx:
Thus
pt- p = p(pt-1- 1) = 0
in [Lt(S2n); Lt(S2n)] and so
p . ss*(Lt(S2n)) = 0
for t > 1. Since L1(S2n) = K(Z(p); 2n), we proves assertion 1.
The proof of assertion 2 is similar to assertion 1, where one needs the fact *
*that the
Samelson product
S2n ^ S2n ! F (S2n)
is of order 2 up to homotopy and the higher Samelson products are null homotopi*
*c_
localized at 2. |*
*__|
Remark 3.14. ss*(L(Sn)Z=p) is known as a specific module over the -algebra [1,
4, 10]. By considering the Bockstein spectral sequence for ss*(L(Sn)), this the*
*orem
shows that ss*(L(S2n) is the kernel of the Bockstein on ss*(L(S2n)Z=p) when n =*
* 1; 3.
Theorem 3.15. Let R = Z(p). If t is not a power of p, then
ss*(Lt(S2n)) = 0:
20 JIE WU
Proof.By a result in [8], there exist functors Amin and Qmaxtfor t 2 such that
1) Amin is a (smallest) coalgebra retract of T with V Amin(V ) for each V ;
2) Qmaxtis a subfunctor of Lt and Qmaxtis a retract of Tt;
3) There is a functorial coalgebra decomposition
M1
T (V ) ~=T ( Qmaxt(V )) Amin(V ):
t=2
Let Lmin(V ) be the primitives of Amin(V ). Then there is a functorial decompos*
*ition
maxM
L(V ) ~=L( Qmax(V )) Lmin(V ):
t=2
Since Qmaxtis a retract of Tt, Qmax(S2n) is a retract of Tt(S2n) and so Qmaxt(S*
*2n)
is either contractible or homotopic to K(Z(p); 2tn). Now Qmaxt(S2n) is contrac*
*tible
because Qmaxtis subfunctor of Lt(V ). It follows that
M1
L( Qmaxt(S2n))
t=2
is contractible. The assertion follows from a result in [8] that Ltis a functor*
*ial retract
of 1
M
L( Qmaxt(S2n))
t=2 *
* __
if t is not a power of p. *
*|__|
By Theorems 3.13 and 3.15, we have
Theorem 3.16. Let {Er}r1 be the integral Adams spectral sequence of F (S1). *
*Then
1) Ert;*= 0 unless t = 2ps some prime p and some non-negative integer s.
2) p . Er2ps;*= 0 for any prime p and any integer s > 0.
3) Let ff 2 E12ps;*with s > 0. Then the only differentials dr, which are pos*
*sibly
non-trivial on ff, are
t-2ps
d2p
with t > s.
References
[1]Bousfield, Curtis, Kan,Quillen, Rector and Schlesinger, The mod-p lower *
*central series and
the Adams spectral sequence, Topology 5 (1966), 331-342.
[2]F. R. Cohen, Cohomology of braid groups, Bull. Amer. Math. Soc. 79 (1973*
*), 761-764.
[3]F. Cohen, On combinatorial group theory in homotopy, Contemp. Math., 188*
* (1995),
57-63.
A BRAIDED SIMPLICIAL GROUP 21
[4]E. B. Curtis, Simplicial homotopy theory, Advancs in Math. (1971), 107-2*
*09.
[5]E. B. Curtis and M. Mahowald, The unstable Adams spectral sequence for S*
*3, Contemp.
Math., 96 (1989), 125-162.
[6]W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Inters*
*cience Publishers
(1966).
[7]J. C. Moore, Homotopie des complexes mon"oideaux, Seminaire Henri Cartan*
* (1954-55).
[8]Paul Selick and Jie Wu, On natural decompositions of loop suspensions an*
*d natural coal-
gebra decompositions of tensor algebras, Memoirs AMS, to appear.
[9]H. Toda, Composition methods in homotopy groups of spheres, Princeton Un*
*iv. Press,
1962.
[10]Robert Wellington, The unstable Adams spectral sequence for free iterate*
*d loop spaces,
Memoirs AMS, 36, no. 258, (1982).
[11]Jie Wu, Combinatorial descriptions of the homotopy groups of certain spa*
*ces, Math. Proc.
Camb. Philos. Soc. to appear.
Department of Mathematics, National University of Singapore, Singapore 117543,
Republic of Singapore, matwuj@nus.edu.sg
~~