On Combinatorial Descriptions of Homotopy
*
Groups of K (ss; 1)
Jie Wu
November 13, 1995
Abstract
We will give a combinatorial description of homotopy groups of
K(ss; 1). In particular, all of the homotopy groups of the 3-sphere
are combinatorially given.
1 Introduction
In this article, we study homotopy groups and some related problems by us-
ing simplicial homotopy theory. The point of view here is that combinatorial
aspects of group theory provide further information about homotopy groups.
The homotopy groups of the 3-sphere, the suspension of K(ss; 1) and
wedges of 2-spheres are shown to be the centers of certain groups with specific
generators and specific relations. We list two group theoretical descriptions
of ss*(S3) as follows.
Definition 1.1 A bracket arrangement of weight n in a group is a set
of elements defined recursively as follows:
Let G be a group and let a1; a2; . .;.an be a finite sequence of elements of
G. Let
fi1(a1) = a1
______________________________
*Research at MSRI is supported in part by NSF grant DMS-9022140
1
and if n > 1, then let
fin(a1; . .;.an) = [fik(a1; . .;.ak); fil(ak+1; . .;.an)]
for some k and l > 0.
Theorem 1.2 For n 1, ssn+2(S3) is isomorphic to the center of the group
with generators y0; : :;:yn and relations
[yffl1i1; yffl2i2; : :;:yffltit]
with {i1; : :;:it} = {-1; 0; : :;:n} as sets in which the indices ij can be re-
peated, where fflj = 1; y-1 = (y0 : :y:n)-1 and the commutator bracket [: :]:
runs over all bracket arrangements of weight t for each t.
Notations 1.3 Let G be a group and let S be a subset of G. Let ~~ denote
the normal subgroup generated by S. Let Hj be a sequence of subgroups of G
for 1 j k. Let [[H1; : :;:Hk]] denote the subgroup of G generated by all of
the commutators [h(1)i1; : :;:h(t)it] with {i1; : :;:it} = {1; : :;:k} as sets *
*in which
the indices ijcan be repeated, where h(s)j2 Hj and the commutator bracket
[: :]:runs over all of the bracket arrangements of weight t for each t.
Theorem 1.4 Let n 1. In the free group F (y0; : :;:yn) freely generated by
y0; : :;:yn
([[; ; : :;:]]\ )=[[; ; : :;:]] ~=ssn+2(S3)
where y-1 = (y0 : :y:n)-1.
The method of the proofs of these theorems is to study the Moore chain
complex of Milnor's construction F (S1) for the 1-sphere S1. A group theo-
retical description of the homotopy groups ss*(K(ss; 1)) is as follows.
Theorem 1.5 Let ss be any group and let {x(ff)|ff 2 J} be a set of generators
for ss. Then, for n 6= 1, ssn+2(K(ss; 1)) is isomorphic to the center of the
quotient group of the free product
groupsa
0jn (ss)j
2
modulo the relations
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit]
with {i1; i2; . .;.it} = {-1; 0; 1; . .;.n} as sets, where (ss)j is a copy of ss
-1 (ff)(ff)-1
with generators {x(ff)j|ff 2 J}, fflj = 1, y(ff)-1= x(ff)0, yj = xj xj+1 for
1 j n - 1, y(ff)n= x(ff)mand the commutator bracket [. .].runs over all of
the commutator bracket arrangment of weight t for each t.
In particular, for ss = Z=m, we have
Corollary 1.6 ssn+2(K(Z=m; 1)) is isomorphic to the center of the group
with generators x0; : :;:xn and relations: (1). xmj for 0 j n and (2).
[yffl1i1; yffl2i2; : :;:yffltit]
with {i1; : :;:it} = {-1; 0; : :;:n} as sets in which the indices ij can be re-
peated, where fflj = 1, y-1 = x-10, yj = xj-1x-1jfor -1 j n - 1,
yn = xn and the commutator bracket [: :]:runs over all bracket arrangements
of weight t for each t.
One of the features of Theorems 1.2 and 1.5 is that homotopy groups
embed in certain 'enveloping groups'. These 'enveloping groups' have sys-
tematic and uniform structure. The centers of these groups are of course
more complicated to analyze. Theorem 1.4 is very similar to the combinato-
rial decription of J. H. C. Whitehead's conjecture [see, Bo2, pp.317]. These
decriptions give a combinatorial question how to give a computable way to
understand the quotient groups R \ S=[R; S] for certain subgroups R and S
of a finite generated free group. Few informations about this question are
known [see, Bo1]. The article is organized as follows.
In Section 2, we study some general properties of simplicial groups. Cen-
tral extensions in the Moore-Postnikov systems will be considered. In Section
3, we study the intersection of certain subgroups in free groups. The proofs
of Theorems 1.2 and 1.4 is given in Section 4, where Theorem 4.5 is Theorem
1.2 and Theorem 4.10 is Theorem 1.4. The proof of Theorem 1.5 is given in
Section 5, where Theorem 5.9 is Theorem 1.5. In Section 6, we give some
applications of our descriptions. One example is to compute the homotopy
groups of the Cohen's construction on the 1-sphere. A direct corollary is to
3
give a short proof of the Milnor's counter example for the minimal simplicial
groups.
The author would like to thank F. Cohen, B. Gray, J. Harper, D. Kan, M.
Mahowald, J. Moore, P. May, S. Priddy and many other mathematicians for
their kindly encouragement and helpful suggestions. The author is indebted
to helpful discussions with J. C. Moore and F. R. Cohen.
2 Central Extensions in simplicial group the-
ory
In this section, we study some general properties of simplicial groups. A sim-
plicial set K is called a Kan complex if it satisfies the extension condition,i*
*.e,
for any simplicial map f : k[n] ! K has an extension g : [n] ! K,
where [n] is the standard n simplex, k[n] is the subcomplex of [n] gen-
erated by all di(oen) for i 6= k and oen is the nondegenerate n simplex in
[n],[K1, C2]. Recall that any simplicial group is a Kan complex [Mo1].
Given aTsimplicial group G, the Moore chain complex (NG; d0) is defined by
NGn = j6=0Kerdj with d0 : NGn ! NGn-1. The classical Moore The-
orem is that ssn(G) ~= Hn(NG) [Mo1, TheoremT4;C2, Theorem 3.7 or K2,
Proposition 5.4]. Now let Zn = Zn(G) = j Kerdj denote the cycles and let
Bn = Bn(G) = d0(NGn+1) denote the boundaries. It is easy to check that
Bn is a normal subgroup of Gn for any simplicial group G.
Lemma 2.1 Let G be a simplicial group. Then the homotopy group ssn(G)
is contained in the center of Gn=BGn for n 1.
Proof. Notice that ssn(G) ~=ZGn=BGn. It suffices to show that the commu-
tator [x; y] 2 BGn for any x 2 ZGn and y 2 Gn.
Notice that x is a cycle. There is a simplicial map fx : Sn ! G such
that fx(oen) = x, where Sn is the standard n-sphere with a nondegenerate n-
simplex oen. Now let the simplicial map fy : [n] ! G be the representative of
y, i.e, fy(on) = y for the nondegenerate n-simplex on. Let OE be the composite
j n [fx;fy]
Sn ! S ^ [n] ! G;
4
where j(oen) = oen ^ on and [fx; fy](a ^ b) = [fx(a); fy(b)], the commutator
of fx(a) and fy(b). Notice that OE(oen) = [x; y] and Sn ^ [n] is contactible.
Thus [x; y] 2 Bn. The assertion follows.
Definitions 2.2 Let G be a simplicial group. The subsimplicial group RnG
is defined by setting
(RnG)q = {x 2 Gq|fx([q])[n]= 1};
where fx is the representative of_x and X[n]is the n-skeleton of the simplicial
set X. The subsimplicial group R nG is defined by setting
__
(R nG)q = {x 2 Gq|fx(([q])n) BGn}:
__ __
Let PnG denote G=RnG and let P nG denote G=R nG.
__
It is easy to check that both_RnG and R nG are normal subsimplicial
group of G. Thus both PnG and P nG are quotient simplicial group of G.
Notice that __
RnG R nG Rn-1G:
__
The quotient simplicial group P nG is between PnG and Pn-1G. By checking
the definition of Moore-Postnikov systems of a simplicial set [Mo1;C2], we
have
Lemma 2.3 The quotient simplicial group PnG is the standard n-th Moore-
Postnikov system of the simplicial group G.
__
The quotient simplicial group P nG has the same homotopy type of PnG.
__
Proposition 2.4 The quotient simplicial homomorphism qn : PnG ! P nG
is a homotopy equivalence for each n.
Proof: The Moore chain complex of PnG is as follows:
8
><1 for q > n + 1,
N(PnG)q = > NGn+1=ZGn+1 for q = n + 1,
: NGq for q < n + 1.
5
__
The Moore chain complex of P nG is as follows:
8
__ ><1 for q > n,
N(P nG)q = > NGn=BGn for q = n,
: NGq for q < n.
__
Thus (qn)* : ss*(PnG) ! ss*(P nG) is an isomorphism and the assertion fol-
lows.
__
__ Let F n denote the kernel of the quotient simplicial homomorphism rn :
PnG ! Pn-1G for n > 0.
__
Proposition 2.5 The simplicial group F n is the minimal simplicial group
K(ssnG; n) for each n > 0.
Proof: It is directly to check that
__ ae1 for q 6= n,
N(F nG)q =
ssnG for q = n.
The assertion follows.
Proposition 2.6 The short exact sequence of simplicial groups
__ __
0 ! F nG ! P nG ! Pn-1G ! 0
is a central extension for each n > 0.
__ __
Proof: Consider_the relative_commutator sussimplicial_group_[F nG; PnG].
By Lemma 2.1, [F nG; PnG]n = 1. Notice_that [F nG; PnG] is a_subsimplicial_
group of minimal simplicial group F nG ~=K(ssnG; n). Thus [F nG; PnG] = 1
and the assertion follows.
Definition 2.7 A simplicial group is said to be r-centerless if the center
Z(Gn) = {1} for n r.
Proposition 2.8 Let G be a reduced r-centerless simplicial group. Then
ssn(G) ~=Z(Gn=Bn) for n r + 1
6
Proof. By Lemma 2.1, Zn=Bn Z(Gn=Bn). It suffices to show that Z(Gn=Bn)
Zn=Bn for n r + 1. Now let "x2 Z(Gn=Bn). Choose x 2 Gn with p(x) = "x,
where p : Gn ! Gn=Bn is the quotient homomorphism. To check "x2 Zn=Bn,
it suffices to show that x 2 Zn or djx = 1 for all j. Since Z(Gn-1) = {1},
djx = 1 if and only if [djx; y] = 1 for all y 2 Gn-1. Now [djx; y] = dj[x; sj-1*
*y]
for j > 0 and [d0x; y] = d0[x; s0y]. Since "x2 Z(Gn=Bn), [x; z] 2 Bn Zn for
all z 2 Gn and therefore [djx; y] = 1 for all y 2 Gn-1. The assertion follows.
By inspecting the proof, we also have
Proposition 2.9 Let G be a reduced r-centerless simplicial group. Then
Z(Gn=Zn) = {1} for n r + 1.
Lemma 2.10 Let G be a reduced simplicial group so that Gn is cyclic or
centerless for each n. Then there exists a unique integer flG > 0 so that
Gn = {1} for n < flG and Z(Gn) = {1} for n > flG .
Proof. Let flG = max{fl|Gn = {1} for n < fl}. Then flG > 0. If flG < 1.
Then GflG6= {1}. We show that GflG+qis centerless for each q > 0. Notice
that dq0O sq0: Gn ! Gn+q ! Gn and dq1O sq1: Gn ! Gn+q ! Gn are identities
for n > 0 Thus sq0(GflG) and sq1(GflG) are nontrivial summands of GflG+q.
Now let x = sq0y = sq1z 2 sq0(GflG) \ sq1(GflG). Then dq+1x = dq+1sq0y =
sq0d1y = 1 = dq+1sq1z = sq-11d2s1z = sq-11z. Thus x = sq1z = 1. And therefore
sq0(GflG) \ sq1(GflG) = {1}. The assertion follows.
Corollary 2.11 Let G be a reduced simplicial group such that Gn is cyclic
or centerless for each n. Then ssn(G) ~=Z(Gn=Bn) for n 6= flG + 1, where flG
is defined as above.
Notice that, for any free group F , rank(F ) 2 , Z(F ) = {1} and
F 6= {1}. We have
Lemma 2.12 Let G be a reduced simplicial group such that Gn is a free
group for each n. Then there exits a unique integer flG > 0 so that Gn = {1}
for n < flG and rank(Gn) 2 for n > flG .
Example 2.13 The 1-stem is determined in this example.
7
Let G = F (Sn), Milnor's F -construction on the standard n-sphere for n 1.
Then Gn ~= F (oe) ~= Z(oe), the free abelian group generated by oe, Gn+1 ~=
F (s0oe; s1oe; : :;:snoe) and Gn+2 ~=F (sisjoe|0 j < i n). It is easy to check
that 2Gn+1 = Zn+1, where qG is the q-th term in the lower central series
of a group G starting with 1G = G. By Lemma 2.1,
3Gn+1 = [Zn+1; Gn+1] Bn+1:
If n = 1, then it will be shown that 3Gn+1 = Bn+1 in Section 4 and
therefore ss3(S2) ~=ss2(F (S1)) ~=Z, which is generated by [s0oe; s1oe].
Suppose that n > 1. Consider the following equations
ae1 for k 6= j;
dk([sj-1sioe; sj+1sjoe]) = [s
ioe; sjoe]for k=j,
for i + 1 < j n,
8
><[si+1oe; si+2oe]for k=i+1,
dk[si+2si+1oe; si+3sioe] = > [si+1oe; sioe]for k=i+3,
: 1 otherwise,
and 8
>><[si+1oe; si+2oe]for k=i+1,
dk[si+2sioe; si+3si+1oe] = > [sioe; si+2oe]for k=i+2,
>:[sioe; si+1oe]for k=i+3,
1 otherwise.
By the Homotopy Addition Theorem [C2, Theorem 2.4], [sioe; sjoe] 2 Bn+1
for i + 1 < j, [si+1oe; si+2oe] [sioe; si+1oe] mod Bn+1 and 0 [sioe; si+2oe]
[si+1oe; si+2oe] 2[sioe; si+1oe] if i + 2 n. Notice that [s0oe; s1oe] 6= 0 in
ssn+1(2G=3G). Thus [s0oe; s1oe] =2Bn+1 and, by the relations above,
ssn+2(Sn+1) ~=ssn+1(G) ~=Zn+1=Bn+1 ~=Z=2
for n 2, which can be represented by [sioe; si+1oe] for 0 i n - 1.
3 Intersections of certain subgroups in free
groups
In this section, we give some group theory preliminary. The intersections of
certain subgroups in free groups are considered in this section. We will use
8
these informations to determine the Moore chain complex of certain simplicial
groups in the next sections.
Definition 3.1 let S be a set and let T S a subset. The projection
homomorphism
ss : F (S) ! F (T )
is defined by ae
ss(x) = x x 2 T ,
1 x =2T .
Now let ss : F (S) ! F (T ) be a projection homomorphism and let R equal
the kernel of ss. Define the subsets of the free group F (S) as follows.
AT(k) = {[[x; yffl11] . .].; yffltt]|0 t k; fflj = 1; y = yffl11:y:f:fltt2 F *
*(T ); x 2 S-T };
where y = yffl11.y.f.fltt2 F (T ) runs over reduced words in F (T ) with t k
and yj 2 T . Furthermore define [[x; yffl11] : :]:; yffltt] = x for t = 0. Defi*
*ne
BT(k) = {OE-1xOE|OE 2 F (T ) a reduced word with lenth l(OE) k; x 2 S - T };
AT = [k0 AT(k)
and
BT = [k0 BT(k):
By the classical Kurosch-Schreier theorem ( see [MKS, pp.243, K2, Theorem
18.1]), we have
Proposition 3.2 The subgroup R is a free group freely generated by BT.
We will show that AT is also a set of free generators for R. We need a
lemma.
Lemma 3.3 Let OE : F1 ! F2 be a homomorphism of free groups. Suppose
that OEab : F1ab! F2abis an isomorphism, where F abis the abelianlizer of the
group F . Then OE : F1 ! F2 is a monomorphism.
9
Proof. Notice that OE* : H*(F1) ! H*(F2) is an isomorphism, where H*(G) is
the homology of the group G. Thus F1=rF1 ! F2=rF2 is an isomorphism
for each r, where rG is the r-th term in the lower central seris of the group
G starting with 1G = G and so
limrF1=rF1 ! limrF2=rF2
is an isomorphism. Notice that \rrF = 1 for any free group F . Thus
F ! limrF=rF is a monomorphism. The assertion follows.
Proposition 3.4 The subgroup R is a free group freely generated by AT.
Proof. First we assume that both S and T are finite sets. Denote by
ik : AT(k) ! R and jk : BT(k) ! R the natural inclusions. Notice that
R = F (BT) = colimkF (BT(k)). We set up the following steps.
Step1. AT(k) F (BT(k)).
The proof of this statement is given by induction on k starting with
AT(0) = BT(0) = S - T . Suppose that AT(k - 1) F (BT(k - 1)) and let
w = [[x; yffl11; . .;.yffltt] 2 AT(k). If t < k, then w 2 AT(k-1) F (BT(k-1))
F (BT(k)), by induction. Now
[[x; yffl11] . .].; yfflkk] = [[x; yffl11] . .].; yfflk-1k-1]-1 . y-fflkk*
*[[x; yffl11] . .].; yfflk-1k-1]yfflkk
*
* Q s -1 j
Now since [[x; yffl11] . .].; yfflk-1k-1] 2 F (BT(k-1)), [[x; yffl11] . .].; yf*
*flk-1k-1] = j=1(OEj xjOEj)) j
with OE-1jxjOEj 2 BT(k - 1) and jj = 1. Thus
Ys Ys
w = ( (OE-1jxjOEj))jj)-1 . (y-fflkkOE-1jxjOEjyfflkk))jj2 F (BT(k))
j=1 j=1
The induction is finished.
Step 2. "ik: F (AT(k)) ! F (BT(k)) is an epimorphism, where the homomor-
phism "ikis induced by the inclusion ik : AT(k) ! F (BT(k))
The proof of this step is given induction on k starting with F (AT(0)) =
F (BT(0)) = F (S - T ). Suppose that F (AT(k - 1)) ! F (BT(k - 1)) is an
10
epimorphism and consider "ik: F (AT(k)) ! F (BT(k)). Let OE-1xOE 2 BT(k),
where OE = yffl11.y.f.flttis a reduced word with t k. If t k - 1, then OE-1xO*
*E 2
Im'k by induction. Let OE = yffl11.y.f.flkkbe a reduced word and let z denote
the word (yffl11.y.f.flk-1k-1)-1xyffl11.y.f.flk-1k-1. Then OE-1xOE = z . [z; yf*
*flkk]. Notice that
z 2 Im("ik) by induction. It suffices to show that [w; yffl] 2 F (AT(k)) for
w 2 AT(k - 1) for all w 2 AT(k - 1), y 2 T and ffl = 1 by the Witt-Hall
identity that
[ab; c] = [a; c] . [[a; c]; b] . [b; c]:
We show this by second induction starting with
AT(1) = {[x; yffl]|y 2 T; x 2 S - T; ffl = 1}:
Let w = [[x; yffl11]; . .].; yffltt] be in AT(k - 1) with k > 1, where yffl11.y*
*.f.flttis
a reduced word. Let y 2 T and let ffl = 1. If yffl11.y.f.flttyfflis a reduced
word, then [w; yffl] 2 F (AT(k)) by definition. Suppose that yffl11.y.f.flttyf*
*flis
not a reduced word. Then t > 0, y = yt and ffl = -fflt. Let w0 denote
[[x; yffl11]; . .;.]; yfflt-1t-1] 2 AT(k - 2). Then w = [w0; yffltt]. By the *
*Witt-Hall
identities, there is an equation
1 = [w0; y-ffltt] . w . [w; y-ffltt]:
By induction, [w0; y-ffltt] 2 F (AT(k - 1)) F (AT(k)) and so [w; y-ffltt] =
w-1[w0; y-ffltt]-1 2 F (AT(k - 1)) F (AT(k)). The second induction is fin-
ished and so the first induction is finished. The assertion follows.
Step 3. "ik: F (AT(k)) ! F (BT(k)) is an isomorphism.
By Step 2, Z(AT(k)) ! Z(BT(k)) is an epimorphism. Notice that
rank(F (BT(k))) = |BT(k)| = |AT(k)| = rank(F (AT(k))):
Thus "ik: Z(AT(k)) ! Z(BT(k)) is an isomorphism and so "ik: F (AT(k)) !
F (BT(k)) is a monomorphism. Thus "ikis an isomorphism.
Step 4. Since F (AT(k)) ! F (BT(k)) is an isomorphism for each k, F (AT) =
colimkF (AT(k)) ! F (BT) = colimkF (BT(k)) is an isomorphism.
11
Now consider the general case. By Lemma 3.6, it suffices to show that
"i: F (AT) ! F (BT) is an isomorphism. To check that F (AT) ! F (BT)
is an epimorphism. Let w 2 BT, there exist finite subsets S0 and T 0of
S and T , respectively, so that w 2 BT0. By the special case as above,
"i|F(A : F (A 0) ! F (B 0) is an isomorphism and w 2 Im"i| . Thus
T0) T T F(AT0)
"iis an epimorphism. To check that F (AT) ! F (BT) is a monomorphism.
Let w 2 Ker"i, there exist finite subsets S0 and T 0of S and T , respectively,
so that w 2 F (AT0). Notice that "i|F(AT0) is an isomorphism. Thus w = 1
and the assertion follows.
Now let's consider the intersection of kernels of projection homomor-
phisms. Let S be a set and let Tj be a subset of S for 1 j k. Let
ssj : F (S) ! F (Tj) be the projection homomorphism for 1 j k. We
construct a subset A(T1; . .;.Tk) of F (S) by induction on k as follows.
A(T1) = AT1;
where AT is defined in Definition3.1. Let
T2(2)= {w 2 A(T1)|w = [[x; yffl11; . .;.yffltt] with x; yj 2 T2 for all *
*j}
and define
A(T1; T2) = A(T1)T(2):
2
Suppose that A(T1; T2; . .;.Tk-1) is well defined so that all of the elemen*
*ts
in A(T1; T2; . .;.Tk-1) are written down as certain commutators in F (S) in
terms of elements in S. Let Tk(k)be the subset of A(T1; T2; . .;.Tk-1) defined
by
Tk(k)= {w 2 A(T1; T2; . .;.Tk-1)|w = [xffl11; . .;.xfflll] with xj 2 Tk for a*
*ll j};
where [xffl11; . .;.xfflll] are the elements in A(T1; T2; . .;.Tk-1) which are*
* written
down as commutators. Then define
A(T1; T2; . .;.Tk) = A(T1; T2; . .;.Tk-1)T(k)
k
Theorem 3.5 Let S be a set and let Tj be a subset of S for 1 j k. Let
ssj : F (S) !TF (Tj) be the projection homomorphism for 1 j k. Then the
intersection kj=1Kerssj is a free group freely generated by A(T1; T2; . .;.T*
*k).
12
Proof. The proof is given by inductionTon k. If k = 1, the assertion follows
from the above lemma. Suppose that k-1j=1Kerssj = F (A(T1; T2; . .;.Tk-1))
and consider ssk : F (S) ! F (Tk). Then
"k
Kerssj = Ker(ssk: F (A(T1; T2; . .;.Tk-1)) ! F (Tk))
j=1
, where sskis ssk restricted to the subgroup F (A(T1; T2; . .;.Tk-1)). Let w =
[xffl11; . .;.xfflll] 2 A(T1; T2; . .;.Tk-1). If w =2Tk(k), then xj =2Tk for so*
*me j and
ssk(w) = 1. Thus sskfactors through F (Tk(k)), i.e, there is a homomorphism
j : F (Tk(k)) ! F (Tk) so that ssk= j O ss, where ss : F (A(T1; T2; . .;.Tk-1))*
* !
F (Tk(k)) is the projection homomorphism. We claim that j : F (Tk(k)) ! F (Tk)
is a monomorphism. Consider the commutative diagram
ssk
F (A(T1; T2; . .;.Tk-1))! F (Tk) ,! F (S)
" " j "
F (Tk(k)) =! F (Tk(k)),! F (A(T1; T2; . .;.Tk-1));
where F (Tk(k)) ! F (A(T1; T2; . .;.Tk-1)) and F (A(T1; T2; . .;.Tk-1)) ! F (S)
are inclusions of subgroups. Thus j : F (Tk(k)) ! F (Tk) is a monomorphism
and
Kerssk= Ker(F (A(T1; T2; . .;.Tk-1)) ! F (Tk(k)) = F (A(T1; T2; . .;.Tk)):
The assertion follows.
Corollary 3.6 Let ssj be the projectionThomomorphisms as in Theorem 3.5.
Then the intersection subgroup kj=1Kerssj equals the commutator subgroup
[[< T1 >; . .;.< Tk >]] which is defined in Notations 1.3.
4 On the Homotopy Groups of the 3-sphere
In this section, we study the Moore chain complex of the Milnor's construc-
tion F (S1). The proofs Theorems 1.2 and 1.4 are given in this section, where
Theorem 4.5 is Theorem 1.2 and Theorem 4.10 is Theorem 1.4. Recall that
13
the simplicial 1-sphere S1 is a free simplicial set generated by a 1-simplex oe.
Thus S10= {*}, S11= {oe; *} and S1n+1= {*; sn . .s.i+1si-1. .s.0oe|0 j n}.
Let xi denote sn . .s.i+1si-1. .s.0oe. Then F (S1)n+1 = F (x0; x1; . .;.xn)
the free group freely generated by x0; . .;.xn. Let yi denote xi-1x-1ifor
-1 i n, where we put x-1 = xn+1 = 1 in F (S1)n+1 = F (x0; . .;.xn). By
direct calculation, we have
Lemma 4.1 F (S1)n+1 = F (y0; . .;.yn) with
8
< yk-1 j k,
djyk = : 1 j = k + 1,
yk j > k + 1,
and 8
< yk+1 j k,
sjyk = : ykyk+1 j = k + 1,and
yk j > k + 1
for 0 j n + 1, where y-1 = (y0 . .y.n-1)-1 in F (S1)n.
Now let Cn+1 denote the subgroup of F (y0; . .;.yn) generated by all of the
commutators [yffl1i1; . .;.yffltit] with {i1; . .;.it} = {0; 1; . .;.n} as sets*
*, i.e. each
j (0 j n) appears in the index set {i1; . .;.it} at least one time, where
fflj = 1 and the commutator [yffl1i1; . .;.yffltit] runs over all of the commut*
*ator
bracket arrangements of weight t for yffl1i1; . .;.yffltit.
Lemma 4.2 The group Cn+1 is a subgroup of NF (S1)n+1,i.e Cn+1 \j6=0Ker(dj).
Proof. Notice that djyj-1 = 1 for 1 j n + 1. Since {yi1; . .;.yit} =
{y0; y1; . .;.yn}, dj[yffl1i1; . .;.yffltit] = 1 for each j > 0. The assertion *
*follows.
Theorem 4.3 NF (S1)n+1 = Cn+1.
Proof. For 1 j n+1, let S = {y0; y1; . .;.yn} and let Tj = {y0; . .;.^yj. .;.*
*yn}.
By Lemma 4.1, there is a commutative diagram
ssj
F (S) ! F (Tj-1)
# dj dj #~=
F (y0; . .;.yn-1)=! F (y0; . .;.yn-1);
14
where ae
dj(yk) = yk 0 k j - 2;
yk-1 j k n:
T n+1 T n+1
Thus Kerdj = Kerssj and NF (S1)n+1 = j=1Kerdj = j=1Kerssj. By
Theorem 3.5, NF (S1)n+1 = F (A(T0; T1; . .;.Tn)), where the notation A(T0; T1; *
*. .;.Tn)
is given in Section 3. To check that
F (A(T0; T1; . .;.Tn)) Cn+1;
it suffices to show that A(T0; T1; . .;.Tn) Cn+1. This will follow from the
following statement.
Statement. For each 0 j n and w = [yffl1i1; yffl2i2; . .;.yffltit] 2 A(T0; T1*
*; . .;.Tj),
{y0; y1; . .;.yj} {yi1; yi2; . .;.yit}.
We show this statement by induction on j. Note that F (T0) = F (y1; . .;.yn).
For j = 0,
A(T0) = {[[y0; yffl1i1]; . .;.]; yffltit]|is > 0; yffl1i1.y.f.fltita reduced *
*word in F (T0)}:
Thus the assertion holds for j = 0. Suppose that the assertion holds for j - 1
with j n. Notice that Tj = {y0; . .;.^yj; . .;.yn}. Thus
Tj(j)= {w 2 A(T0; T1; . .;.Tj-1)|w = [yffl1i1; . .;.yffltit] with yj =2{yi1; .*
* .;.yit}}:
and so yj 2 {yi1; . .;.yit} for w = [yffl1i1; . .;.yffltit] 2 A(T0; T1; . .;.Tj*
*-1) - Tj(j).
Hence, by induction,
{y0; y1; . .;.yj} {yi1; yi2; . .;.yit}
for w = [yffl1i1; . .;.yffltit] 2 A(T0; T1; . .;.Tj-1) - Tj(j). Notice that
A(T0; T1; . .;.Tj) = A(T0; T1; . .;.Tj-1)T(j):
j
Thus
{y0; y1; . .;.yj} {yi1; yi2; . .;.yit}
for each w = [yffl1i1; . .;.yffltit] 2 A(T0; T1; . .;.Tj). The induction is fin*
*ished and
the theorem follows.
15
Corollary 4.4 A(T0; T1; . .;.Tn) is a set of free generators for NF (S1)n+1.
Theorem 4.5 (Theorem 1.2) For n 1, ssn+2(S3) is isomorphic to the
center of the group with generators y0; y1; . .;.yn and relations [yffl1i1; yff*
*l2i2; . .;.yffltit]
with
{i1; i2; . .;.it} = {-1; 0; 1; . .;.n}
as sets, where fflj = 1,y-1 = (y0y1 . .y.n)-1 and the commutator bracket [. .].
runs over all bracket arrangements of weight t for each t.
Proof. Notice that ssn+2(S3) ~=ssn+1F (S1) for n 1 and Bn+1 = d0(NF (S1)n+2).
By Lemma 4.1 and Theorem 4.3, Bn+1 is generated by [yffl1i1; yffl2i2; . .;.yffl*
*tit] so that
{yi1; yi2; . .;.yit} = {y0; y1; . .;.yn}, where fflj = 1. By Proposition 2.8, *
*it
suffices to check that ss2(F (S1)) ~= Z(F (S1)2=B2) = Z(F (y0; y1)=B2). By
Example 2.13, Z2 = 2F (S1)2 = 2F (y0; y1) and 3F (y0; y1) B2. Now ,
by the construction of B2, B2 3F (y0; y1) and therefore
ss2(F (S1)) ~=Z2=B2 = 2F (y0; y1)=3F (y0; y1) = Z(F (y0; y1)=B2):
The assertion follows.
Remark 4.6 The relations in above theorem are not minimal, i.e, many of
them can be cancelled out.
Proposition 4.7 Let C0 n+1 be the subgroup of F (S1)n+1 generated by all
commutators given by [. .[.yffl1i1; yffl2i2]; . .;.]; yffltit] with {i1; . .;.i*
*t} = {0; 1; . .;.n}
and fflj = 1. Then
NF (S1)n+1=s \ NF (S1)n+1 ~=C0n+1=s \ C0n+1
for each s, where s = sF (S1)n+1 is the s-term in the lower central series
of F (S1)n+1.
Proof. Notice that C0n+1 NF (S1)n+1. The induced homomorphism fs :
C0n+1=s \ C0n+1! NF (S1)n+1=s \ NF (S1)n+1 is a monomorphism. We
check that f is an epimorphism. It suffices to show that, for each w 2
NF (S1), there exists a sequence of elements {xj} so that xj 2 j\ C0n+1and
wx1x2 . .x.s2 s+1 for each s. In fact, if this statement holds, wx1x2 . .x.s-1
1 mod s\ NF (S1)n+1 for each s and w = (wx1 . .x.s-1) . (x1x2 . .x.j-1)-1 2
16
C0n+1mod s. Now we construct xj by induction, which depends on w.
Notice that, for n 1, NF (S1)n+1 2 n+1 2. Choose xj = 1 for
j n. Suppose that there are x1; . .;.xs-1 so that xj 2 j \ C0n+1and
wx1 . .x.s-12 s. Since C0n+1 NF (S1)n+1, wx1 . .x.s-12 s \ NF (S1)n+1
and dj(wx1 . .x.s-1) = 1 for j > 1. Let ss : s ! s=s+1 be the quotient ho-
momorphism. ss(wx1 . .x.s-1) is a linear combination of basic Lie products.
We claim that ss(wx1 . .x.s-1) is a linear combination of basic Lie products in
which each yj appears in the Lie product for 0 j n. If not, there exists
j so that ss(wx1 . .x.s-1) = b + c, where b is a nontrivial linear combination
of basic Lie products in which yj does not appear and c is a linear combina-
tion of basic Lie products in which yj appears. Now the face homomorphism
dj+1 : F (y0; . .;.yn) ! F (y0; . .;.yn-1) induces a homomorphism
dj+1: sF (y0; . .;.yn)=s+1F (y0; . .;.yn) ! sF (y0; . .;.yn-1)=s+1F (y0; . .;.y*
*n-1)
and
1 = dj+1ss(wx1 . .x.s-1) = dj+1(b) + dj+1(c) = dj+1(b):
Notice that
dj+1|F(y0;...;yj-1;yj+1;...;yn): F (y0; . .;.yj-1; yj+1; . .;.yn) ! F (y0; . *
*.;.yn-1)
is an isomorphism. Thus b = 1. This contradicts to that b is a nontriv-
ial linear combination of a basis. Thus ss(wx1 . .x.s-1) is a linear combi-
nation of basic Lie products in which all of the yj appear. By Theorem
5.12 in [MKS,pp.337], there exists xs in C0n+1so that ss(wx1 . .x.s) = 1, or
wx1 . .x.s2 s+1. The induction is finished now.
By Corollary 3.6, we have
Theorem 4.8 In the free group F (y0; . .;.yn), NF (S1)n+1 = [[< y0 >; . .;.<
yn >]] and BF (S1)n+1 = [[< y-1 >; < y0 >; . .;.< yn >]].
Thus Theorem 4.5 can be rewritten as follows.
Theorem 4.9 In the free group F (y0; . .;.yn) for n 1, the center
Z(F (y0; . .;.yn)=[[< y-1 >; < y0 >; . .;.< yn >]]) ~=ssn+2(S3)
17
By Lemma 4.1 and Proposition 3.5, Kerd0 =< x0 >=< y-1 > and
therefore
Zn+1 = [[< y0 >; . .;.< yn >]]\ < y-1 > :
Thus we have
Theorem 4.10 (Theorem 1.4) In the free group F (y0; . .;.yn) with n 1,
[[< y0 >; . .;.< yn >]]\ < y-1 >]]=[[< y-1 >; < y0 >; . .;.< yn >]] ~=ssn+2(S3):
5 On the Homotopy Groups of K(ss; 1)
In this section, we give group theoretical descriptions for ss*(K(ss; 1)) for
any group ss. The proof of Theorem 1.5 is given in this section, where The-
orem 5.9 is Theorem 1.5. We will use the notations defined in Section 3.
First we extend our description for ss*(S2) to the case ss*(_ff2JS2). Re-
call that (_ff2JS1)0 = *, (_ff2JS1)1 = {oeff; *|ff 2 J} and (_ff2JS1)n+1 =
{sn . .^.si.s.0.oeff; *|ff 2 J; 0 i n}. Let x(ff)idenote sn . .^.si.s.0.oeff.
Then
F (_ff2JS1)n+1 = F (x(ff)0; x(ff)1; . .;.x(ff)n|ff 2 J):
Let y(ff)j= x(ff)j. x(ff)j+1for 0 j n - 1 and y(ff)n= x(ff)n. By Lemma 4.1, we
have
Lemma 5.1 F (_ff2JS1)n+1 = F (y(ff)j|0 j n; ff 2 J) with
8
> 1 j = k + 1,
: (ff)
yk j > k + 1,
and 8
>> y(ff). y(ff)j = k + 1,
>: k(ff) k+1
yk j > k + 1,
for 0 j n + 1, where y(ff)-1= (y(ff)0.y.(.ff)n-1)-1 in F (_ff2JS1)
Let CJn+1denote the subgroup of F (_ff2JS1)n+1 generated by all of the
commutators [y(ff1)ffl1i1; . .;.y(fft)ffltit] with {i1; . .;.it} = {0; 1; . .;.*
*n} as sets, where
fflj = 1, ffj 2 J and the commutator bracket [. .].runs over all of the com-
mutator bracket arrangements of weight t for each t.
18
Lemma 5.2 CJn+1 NF (_ff2JS1)n+1.
Proof: For each 1 j n+1, there exists some is = j -1. Thus dj(y(ffs)fflsis) =
1 for some is and therefore
dj([y(ff1)ffl1i1; . .;.y(fft)ffltit]) = 1
for each j > 0. The assertion follows.
Lemma 5.3 For each 1 j n + 1,
Kerdj \ F (_ff2JS1)n+1 =< y(ff)j-1|ff 2 J >;
the normal subgroup generated by y(ff)j-1with ff 2 J.
Proof: By the definition of dj, there is a commutative diagram
p (ff) (ff) (ff)
F (y(ff)j|0 j n; ff 2 J) ! F (y0 . .^.yj-1.y.n.|ff 2 J)
dj # ~=# dj
F (y(ff)j|0 j n - 1; ff 2 J)=! F (y(ff)j|0 j n - 1; ff 2 J)
where p is the projection and
( (ff)
djy(ff)k= yk-1 j k,
y(ff)kj > k + 1.
The assertion follows.
Theorem 5.4 Let CJn+1be defined as above. Then
NF (_ff2JS1)n+1 = CJn+1
Proof: By lemma 5.1, each dj with j > 0 is a projection homomorphism.
Thus, by Theorem 3.5,
n+1"
NF (_ff2JS1)n+1 = Kerdj = F (A(T0; T1; . .;.Tn))
j=1
where Tj = {y(ff)0; . .;.^y(ff)j; . .;.y(ff)n|ff 2 J}. It suffices to show that*
* A(T0; T1; . .;.Tn)
CJn+1. This follows from the next lemma.
19
Lemma 5.5 For 0 j n, let W = [y(ff1)ffl1i1; . .;.y(fft)ffltit] 2 A(T0; T1;*
* . .;.Tj).
Then
{0; 1; . .;.j} {i1; . .;.it}:
Proof: The proof is given by induction on j for 0 j n. Notice that,
by construction, each element in A(T0; T1; . .;.Tj) is written as a certain
commutator. If j = 0, then
A(T0) = {y(ff)0; [[y(ff)0; y(ff1)ffl1i1]; . .;.]; y(fft)ffltit]}
where ff; ffj 2 J, fflj = 1 and y(ff1)ffl1i1.y.(.fft)ffltitruns over all of the*
* reduced
words 6= 1 in F (y(ff)j|ff 2 J; 1 j n). Thus the assertion holds for j = 0.
Suppose that the the assertion holds for j - 1 with j n. Recall that
Tj(j)= {W 2 A(T0; T1; . .;.Tj-1|W = [y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffl*
*tit]; with y(ffs)is2 Tj}:
Notice that y(ffs)is2 Tj () is 6= j. Thus, for W = [y(ff1)ffl1i1; y(ff2)ffl2i2;*
* . .;.y(fft)ffltit] 2
A(T0; . .;.Tj-1) - Tj(j), j 2 {i1; . .;.it}. By induction, {0; . .;.j - 1}
{i1; . .;.it}. Hence
{0; 1; . .;.j} {i1; i2; . .;.it}
for any W = [y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit] 2 A(T0; . .;.Tj-1 -*
* Tj(j). Recall that,
by construction,
A(T0; . .;.Tj) = AT(j):
j
{0; 1; . .;.j} {i1; i2; . .;.it}
for any W = [y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit] 2 A(T0; . .;.Tj). T*
*his completes the
proof.
Theorem 5.6 ssn+2(_ff2JS2) is isomorphic to the center of the group with
generators
y(ff)0; y(ff)1; . .;.y(ff)n
for ff 2 J. and relations
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit]
with {i1; i2; . .;.it} = {-1; 0; 1; . .;.n} as sets, where the indices ij can be
repeated, fflj = 1, ffj 2 J and the commutator bracket runs over all of the
commutator bracket arrangements of weight t for each t.
20
Proof: Notice that ssn+2(_ff2JS2) ~= ssn+1(F (_ff2JS1)). By the above theo-
rem, Bn+1 is generated by
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit]
By Theorem 2.8, the assertion holds for n 1. For n = 0, B1 = 2(F (y(ff)0|ff 2
J)) and
F (y(ff)0|ff 2 J)=B1 ~=ff2JZss2(_ff2JS2):
The assertion holds for all of n.
For the general case, we need a simplicial group construction.
Definition 5.7 Let G be a simplicial group and let X be a pointed simplicial
set with a point *. The simplicial group F G(X) is defined by setting
a
F G(X)n = (Gn)x;
x2Xn
the free product, modulo the relations (Gn)*, where (Gn)x is a copy of Gn. The
faces and degeneracies homomorphisms in F G(X) is given in the canonical
way by the universal property of the coproduct in the category of groups and
group homomorphisms.
Lemma 5.8 (Ca, Theorem 9, pp.88) Let G be a simplicial group and let
X be a pointed simplicial set. Then the goemetric realization |F G(X)| is
homotopy equivalent to (B|G| ^ |X|).
A generalization of this lemma by using fibrewise simplicial groups is
given in [Wu1].
Theorem 5.9 (Theorem 1.5) Let ss be any group and let {x(ff)|ff 2 J} be
a set of generators for ss. Then, for n 6= 1, ssn+2(K(ss; 1)) is isomorphic to
the center of the quotient group of the free product
groupsa
0jn (ss)j
modulo the relations
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit]
21
with {i1; i2; . .;.it} = {-1; 0; 1; . .;.n} as sets, where (ss)j is a copy of ss
-1 (ff)(ff)-1
with generators {x(ff)j|ff 2 J}, fflj = 1, y(ff)-1= x(ff)0, yj = xj xj+1 for
1 j n - 1 and y(ff)n= x(ff)n, the commutator bracket [. .].runs over all of
the commutator bracket arrangment of weight t for each t.
Proof: Since {x(ff)|ff 2 J} is a set of generators for ss, F (x(ff)|ff 2 J) ! s*
*s is
an epimorphism. Thus
(ff)|ff2J)1 ss 1
F (_ff2JS1) ~=F F(x (S ) ! F (S )
is an epimorphism. Hence
NF (_ff2JS1) ! NF ss(S1)
and
BF (_ff2JS1) ! BF ss(S1)
are epimorphisms. The assertion follows from Theorem 5.6, Lemma 5.7 and
Proposition 2.8.
6 Applications
In this section, we consider Cohen's K-construction. Our descriptions for
the homotopy groups of 3-sphere gives a calculation for the K-construction
of 1-sphere.
Definition 6.1 Let X be set. The group K(X) is defined to the the quotient
group of the free group F (X) modulo the normal subgroup generated by all
of the commutators [[x1; x2]; . .;.]; xt] with xi 2 X and xi = xj for some
1 i < j t Now let S be a pointed simplicial set. The simplicial group
K(S) is defined to be the quotient simplicial group of F (S) modulo the normal
simplicial subgroup generated by all of the commutators [[x1; x2]; . .;.]; xt] *
*with
xi 2 S and xi = xj for some 1 i < j t, where F (S) is the Milnor's
F (K)-construction for the simplicial set S.
Definition 6.2 The group Lie(n) is the elments of weight n in the Lie alge-
bra Lie(x1; x2; . .;.xn) which is the quotient Lie algebra of the free Lie alge*
*bra
L(x1; x2; . .;.xn) over Z modulo the two sided Lie ideal generated by the Lie
elements [[xi1; xi2]; . .;.]; xit] with il= ik for some 1 l < k t.
22
The following lemmas are due to Fred Cohen.
Lemma 6.3 (Co) qK(x1; x2; . .;.xn) = {1} for q n and
nK(x1; x2; . .;.xn) ~=Lie(n);
where qG is the q-th term in the lower central series for the group G starting
with 1G = G.
Lemma 6.4 (Co) In the group K(x1; x2; . .;.xn), the normal subgroup grn-
erated by xj is abelian for each 1 j n.
Lemma 6.5 (Co) The set {[x1; xoe(2); . .;.xoe(n)]|oe 2 n-1} is a Z-basis for
Lie(n), where n-1 acts on {2; 3; . .;.n} by permutation.
Recall that the simplicial 1-sphere S1 is a free simplicial set generated
by a 1-simplex oe. More precisely, S10 = {*}, S11 = {oe; *} and S1n+1 =
{*; sn . .^.sj.s.0.oe|0 j n}. Let xi denote sn . .^.sj.s.0.oe. Then
Lemma 6.6 The face functions di : S1n+1! S1nand the degenarate functions
si : S1n+1! S1n+2are as follows:
aex j < i
dixj = xj
j-1 j i
and ae
xj j < i
sixj = x ;
j+1 j i
where we put x-1 = * and xn = * in S1n.
Theorem 6.7 ssn(K(S1)) is isomorphic to Lie(n)
Proof: Let ss : F (S1) ! K(S1) be the quotient homomorphism. Then
NF (S1) ! NK(S1) is an epimorphism. Recall that NF (S1)n+1 is generated
by all of the commutators
[yi1; yi2; . .;.yit]
so that {i1; i2; . .;.it} = {0; 1; . .;.n} by Theorem 4.3. Thus NF (S1)n+1
n+1F (S1)n+1 and therefore
NK(S1)n+1 n+1K(S1)n+1:
23
Notice that K(S1)n+1 ~=K(x0; x1; . .;.xn). Thus n+1K(S1)n+1 ~=Lie(n+1)
and n+1K(S1)n = {1}. Thus
dj|n+1K(S1)n+1 : n+1K(S1)n+1 ! K(S1)n
is trivial for each j 0. And therefore
NKS1)n+1 = n+1K(S1)n+1 ~=Lie(n + 1)
with d0 : NK(S1)n+1 ! NK(S1)n is trivial. the assertion follows.
Corollary 6.8 Let ss : F (S1) ! K(S1) be the quotient simplicial homomor-
phism. Then
ss* : ssn(F (S1)) ! ssn(K(S1))
is an isomorphism for n = 1; 2 and zero for n > 2.
Now we consider the Samelson product in ss*(K(S1)). Let xj denote sn . .^.sj.s.*
*0.oe
in S1n+1. The following lemma follows directly from Lemma 6.6.
Lemma 6.9 Let I = (i1; i2; . .;.im ) be a sequence with i1 < i2 < . .<.im .
Then sI : S1n+1- * ! S1n+m+1- * is the composite
{x0; x1; . .;.xn} sI!{x0; x1; . .;.^xi1; . .;.^xi2; . .;.^xim; . .;.xn+m*
* }
,! {x0; x1; . .;.xn+m }:
where sI = sim . .s.i1and sI is the order preserving isomorphism.
Recall that, for oe 2 ssn(G) and o 2 ssm (G), the Samelson product [C1] is
defined to be Y
< oe; o >= [sboe; sao]sign(a;b);
(a;b)
where G is a simplicial group, (a; b) = (a1; . .;.an; b1; . .;.bm ) runs over a*
*ll
shuffles of (0; 1; . .;.m + n - 1),i.e. all permutations, so that a1 < a2 < . .*
*<.
an, b1 < b2 < . .<.bmQ, sign(a; b) is the sign of the permutation (a; b), the
order of the product is right lexicographic on a and sa = san . .s.a1.
24
Proposition 6.10 The Samelson product in ss*(K(S1)) is as follows:
< [xoe(0); xoe(1); . .;.xoe(n)]; [xo(0); xo(1); . .;.xo(m)] >
X
= sign(I; J)[[xioe(0); . .;.xioe(n)]; [xjo(0); . .;.xjo(m)]]
(I;J)
for the commutators
[xoe(0); xoe(1); . .;.xoe(n)] 2 ssn+1(KS1)) ~=Lie(n + 1)
and
[xo(0); xo(1); . .;.xo(m)] 2 ssm+1 (K(S1)) ~=Lie(m + 1)
where
(I; J) = (i0; i1; . .;.in; j0; j1; . .;.jm )
runs over all shuffles of (0; 1; . .;.m + n + 1) so that i0 < i1 < . . .< in,
j0 < j1 < . .<.jm , sign(I; J) is the sign of the permutation (I; J), oe 2 n+1
acts on {0; 1; . .;.n} and o 2 m+1 acts on {0; 1; . .;.m}
Proof: Notice that {x0; . .;.^xj0; . .;.^xjm; . .;.xn+m+1 } = {xi0; . .;.xin} *
*and
sJ: {x0; . .;.xn} ! {xi0; . .;.xin} is an ordered isomorphism.
sJ([xoe(0); xoe(1); . .;.xoe(n)]) = [xioe(0); . .;.xioe(n)]
and
sI([xo(0); xo(1); . .;.xo(m)]) = [xjo(0); . .;.xjo(m)]:
The assertion follows.
Definition 6.11 A simplicial group is minimal if it is also a minimal sim-
plicial set.
Recall that a simplicial group G is minimal if and only if the Moore chain
complex NG is minimal [C2].
Theorem 6.12 The simplicial group K(S1) is the universal minimal sim-
plicial quotient simplicial group of F (S1) in the following sense:
(1). K(S1) is a minimal simplicial group.
(2). Let G be a minimal simplicial group. Then every simplicial homomor-
phism f : F (S1) ! G factors through K(S1).
25
Proof: By inspecting the proof of Theorem 6.7, K(S1) is a minimal simplicial
group. The assertion (2) follows from the following statement.
Statement: K(S1) is the quotient simplicial group of F (S1) modulo the nor-
mal subsimplicial group generated by the boundaries.
__ Let H denote the kernel of the quotient map p : F (S1) ! K(S1) and let
B denote the normal subsimplicial group of F (S1) generated by the bound-_
aries BF (S1). Notice that K(S1) is a minimal simplicial group. Thus_B_
is contained in H. Let Q denote the quotient simplicial group F (S1)=B .
Then Q is a minimal simplicial group. By Proposition 2.6, there is a central
extension
0 ! K(ssn+1Q; n + 1) ! Pn+1Q ! PnQ ! 0;
where PnQ is the n-th Moore-Postnikov system of Q. Notice that P1Q =
K(ssQ ; 1) = K(Z; 1). Thus n+2Pn+1Q = 1 by induction on n. Notice that
Qn+1_~=(Pn+1Q)n+1. Thus n+2Qn+1 = 1. Now we show that H_is_contained
in B by induction_on the dimension starting with H1 = B 1 = 1. Suppose
that Hn B n with n > 0. Notice that F (S1)n+1 = F (x0; . .;.xn) and
K(S1)n+1 = K(x0; . .;.xn). Thus Hn+1 is a normal subgroup of F (x0; . .;.xn)
generated by the commutators
[[xi1; xi2]; . .;.xit]; xit]
such that ip 6= iq for p < q. Now consider W =_[[xi1; xi2]; . .;.xit]; xit]. *
*If
t n+1, then W 2 n+2F (x0; . .;.xn). Thus W 2 B n+1since n+2Qn+1 = 1.
If t < n + 1, then there exists an index j 2 {0; 1; . .;.n} - {i1; . .;.it}.
Recall that ae
xk k < i,
sixk =
xk+1 k i,
for xk = sn-1 . .^.sk.s.0.oe 2 S1n. Thus
sj[[xi01; xi02]; . .;.xi0t]; xi0t] = [[xi1; xi2]; . .;.xit]; xit];
where_i0k= ik if ik < j and i0k= ik-1 if ik_>_j. By induction, [[xi01; xi02]; .*
* .;.xi0t]; xi0t] 2
B. Thus W = [[xi1; xi2]; . .;.xit]; xit] 2 B . The induction is finished and the
assertion follows.
The simplicial group K(S1) is homotopy eqivalent to a product of the
Eilenberg-Maclane spaces with a different product structure.
26
Proposition 6.13 K(S1) is an abelian simplicial group. Therefore K(S1)
is homotopy equivalent to a product of the Eilenberg-MacLane spaces as a
simplicial set.
Proof: Consider
d0 : K(x0; x1; . .;.xn) ! K(x0; x1; . .;.xn-1)
d0(x0) = 1 and d0(xj) = xj-1: Thus Kerd0 \ Kn+1(S1) ~=< x0 > is the nor-
mal subgroup generated by x0 which is abelian by Lemma 6.4. The assertion
follows.
In the end of this section, we give some applications of K(S1) to minimal
simplicial groups.
Proposition 6.14 Let G be a minimal simplicial group such that the abelian-
lizer Gab is a minimal simplicial group K(ss; 1) for a cyclic group ss. Then G
is homotopy equivalent to a product of Eilenberg-Maclane spaces.
Proof: Notice that G1 = ss. Let x be a generator for the cyclic group ss and
let fx : S1 ! G be a representive map of x, i.e, fx(oe) = x. Let g : F (S1) ! G
be the simplicial homomorphism induced by fx. We need a lemma.
Lemma 6.15 The simplicial homomorphism g : F (S1) ! G is simplicial
surjection.
Proof: It suffices to show that the subsimplicial group, denote by H, of G
generated by G1 is G itself. This is given by induction on the dimensions
starting with H1 = G1. Suppose that Hn-1 = Gn-1 with n > 1. By a result
of Condule [see, e.g, Po, Proposition 1, pp.6], Gn is generated by the de-
generate images of lower order Moore chain complex terms and NGn. Thus
Gn is genereted by NGn and Hn by induction. Notice that G is a minimal
simplicial group. Thus NGn = ZGn, the cycles. By Proposition 2.1, ZGn
is contained in the center of Gn. Thus Hn is a normal subgroup of Gn and
the composite OE : ssnG ~= ZGn ! Gn ! Gn=Hn is an epimorphism. Thus
Gn=Hn is an abelian group and so the quotient homomorphism Gn ! Gn=Hn
factors through Gabn. Notice that Gab = K(ss; 1). Thus Gabn~=(Gn=ZGn)ab
and so OE : ssnG ! Gn=Hn is trivial. Thus Gn=Hn is trivial and the assertion
follows.
27
Continuation of Proof of Proposition 6.14: Notice that G is minimal. The
simplicial epimorphism g : F (S1) ! G factors through K(S1) by Proposi-
tion 6.12. By Proposition 6.13, K(S1) is an abelian simplicial group. Thus
G is also an abelian simplicial group. Thus G is homotopy equivalent to a
product of Eilenberg-Maclane spaces, which is the assertion.
The following counter-example for minimal simplicial groups is due to J.
W. Milnor (unpublished).
Proposition 6.16 (Sn+1[n + 1; n + 2; n + 3]) does not have a homotopy
type of a minimal simplicial group for n > 0, where Sn+1[n + 1; n + 2; n + 3]
is the 3-stage Postnikov system by taking the first three nontrivial homotopy
groups of Sn+1.
Proof: Suppose that G is a minimal simplicial group such that G ' (Sn+1[n+
1; n + 2; n + 3]). Let f : F (S1) ! n-1G be a simplicial homomorphism such
that f(oe) is a generator of (n-1G)1 ~= Gn ~= Z. Then f* : ssj(F (S1)) !
ssj(n-1G) is an isomorphism for j 3. Notice that n-1G is also a min-
imal simplicial group. The simplicial homomorphism f : F (S1) ! n-1G
factors through K(S1). Notice that ss3(F (S1)) ~= ss3n-1G) ~= Z=2 and
ss3(K(S1)) ~= Lie(3) ~= Z Z. There is a contradiction and the assertion
follows.
More examples and counter-examples for minimal simplicial groups will
be given in [Wu2]. It was know that there are still many counter-examples
of two-stage Postnikov systems for minimal simplicial groups [Wu2].
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