ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS
OF CERTAIN SPACES
JIE WU
Abstract. We will give a combinatorial description of the homotopy groups*
* for
the suspension of K(ss; 1) and wedges of 2-spheres. In particular, all of*
* the homotopy
groups of the 2-sphere are given as the centers of certain combinatoriall*
*y described
groups.
1. Introduction
In this article, we study homotopy groups and some related problems by using
simplicial homotopy theory. The point of view here is that combinatorial aspect*
*s of
group theory provide further information about homotopy groups. This view does
not yet admit computational applications to higher homotopy groups, but it does
provide a somewhat different approach than those given classically. In particu*
*lar,
in this paper, homotopy groups are given as centers of groups which admit natur*
*al
combinatorial descriptions.
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 generator*
*s and
specific relations. We list two group theoretical descriptions of ss*(S3) as fo*
*llows. Let
G be a group and let [x; y] = x-1y-1xy in G.
Definition 1.1. A bracket arrangement of weight n in a group G is a map
fin :Gn ! G satisfying the following recursive property:
fi1 = idG
fin(a1; . .;.an) = [fik(a1; . .;.ak); fil(ak+1; . .;.an)]
for some k + l = n, 0 < k; l < n.
Consider the free group with a fixed choice of generators F (y0; y1; . .;.yn)*
*. Let Bn
denote the subgroup of F (y0; y1; . .;.yn) generated by the commutators
fit(xffl11; xffl22; . .;.xffltt);
where
___________
Research at MSRI is supported in part by NSF grant DMS-9022140.
1
2 JIE WU
1) fflj = 1;
2) xs 2 {y-1; y0; y1; . .;.yn} with y-1 = y0y1y2. .y.n;
3) all elements in {y-1; y0; y1; . .;.yn} appear as at least one of the xj;
4) for each t n + 2, fit runs over all of the commutator bracket arrangemen*
*ts
of weight t.
Notice that Bn is a normal subgroup of F (y0; y1; . .;.yn).
Theorem 1.2. For n 1, ssn+2(S3) is isomorphic to the center of F (y0; y1; . *
*.;.yn)=Bn.
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 commuta*
*tors
fit(h(1)i1; : :;:h(t)it), where
1) 1 is k;
2) all integers in {1; 2; . .;.k} appear as at least one of the integers is;
3) h(s)j2 Hj;
4) for each t k, fit runs over all of the commutator bracket arrangements of
weight t.
Theorem 1.4. Let n 1 and let F (y0; : :;:yn) be the free group generated by *
*y0; : :;:yn.
Then there is an isomorphism of groups
[[;_;_:_:;:]]_\~3
= ssn+2(S )
[[; ; : :;:]]
where y-1 = y0: :y:n.
The method of the proofs of these theorems is to study the Moore chain comple*
*x of
Milnor's F (K)-construction [18] for the 1-sphere S1 and to use the modified Mo*
*ore-
Postnikov system (Section 2). A group theoretical description of the homotopy g*
*roups
ss*(K(ss; 1)) for general ss is as follows.
Theorem 1.5. Let ss be any group and let {x(ff)|ff 2 J} be a set of generator*
*s for ss.
Let (ss)j be a copy of ss with generators {x(ff)j|ff 2 J}. Then, for n 6= 1, ss*
*n+2(K(ss; 1))
is isomorphic to the center of the quotient group of the free product
groupsa
(ss)j
0jn
modulo the relations
fit(y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit);
where
1) fflj = 1;
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 3
2) -1 is n;
3) all integers in {-1; 0; 1; 2; . .;.n} appear as at least one of the integ*
*ers is;
-1 (ff)(ff)-1 (ff) (ff)
4) y(ff)-1= x(ff)0, yj = xj xj+1 for 1 j n - 1 and yn = xn ;
5) for each t n + 2, fit runs over all of the commutator bracket arrangemen*
*t of
weight 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:
I) xmjfor 0 j n and
II)fit(yffl1i1; yffl2i2; : :;:yffltit);
where
1) ffl = 1;
2) -1 is n;
3) all integers in {-1; 0; 1; 2; . .;.n} appear as at least one of the integ*
*ers is;
4) y-1 = x-10, yj = xj-1x-1jfor -1 j n - 1 and yn = xn;
5) for each t n + 2, fit runs over all bracket arrangements of weight t.
Theorem 1.5 is general in the following sense. By the Kan-Thurston Theorem [1*
*6],
for any connected space X, there exist a group ss and a map f :K(ss; 1) ! X such
that
f*: H*(K(ss; 1)) ! H*(X)
is an isomorphism, that is, f is a homology equivalence. Thus f :K(ss; 1) ! X is
a (weak) homotopy equivalence. Hence Theorem 1.5 gives a combinatorial descript*
*ion
of the homotopy groups of any suspension.
One of the features of Theorems 1.2 and 1.5 is that homotopy groups embed in
certain `enveloping groups'. These `enveloping groups' have systematic and unif*
*orm
structure. The centers of these groups are of course more complicated to analy*
*ze.
Theorems 1.2 and 1.4 are the first descriptions, but not explicit calculations,*
* of the
general homotopy groups of the 3-sphere. These descriptions give new informati*
*on
about the homotopy groups of the 3-sphere. Also these descriptions allow us to
calculate the homotopy groups of Cohen's K-construction of the 1-sphere.
The homotopy groups of K(ss; 1) have been studied by many people from dif-
ferent point of view (see, for examples, [2, 3, 4, 10, 11, 12, 13, 26, 24]). Br*
*own and
Loday [3] gave a complete description of ss3(K(ss; 1)), which leads to calculat*
*ions
of ss3(K(ss; 1)). Theorem 1.5 gives a description of the general higher homoto*
*py
groups. So far calculations on the higher homotopy groups are still out of rea*
*ch,
4 JIE WU
but the combination of these different techniques for describing the higher hom*
*o-
topy groups has been pursued [22] after Theorem 1.5 was first announced in auth*
*or's
thesis.
We emphasize that descriptions here are NOT calculations. In Theorem 1.2, we
give an explicit group which occurs naturally for which the homotopy group of t*
*he 3-
sphere is the center. These structures do not immediately give any new or infor*
*mative
calculation on the higher homotopy groups in the classical sense, but they give*
* a
different view of the underlying structure that occurs in a natural way. It is*
* an
attempt to consider these groups in a systematic way.
Theorem 2.20 states that the torsion component of any homotopy group of a sim*
*ply
connected space is the torsion component of the center of certain nilpotent gro*
*up. In
Proposition 4.9, we give an explicit finitely presented nilpotent group for whi*
*ch the
higher homotopy group of the 3-sphere is the torsion component of the center.
There is certain similarity between the combinatorial description of the homo*
*topy
groups of the 2-sphere given in Theorem 1.4 and the combinatorial description o*
*f J.
H. C. Whitehead's conjecture given in [5]. Notice that the homotopy groups of t*
*he
2-sphere and the Whitehead conjecture are different problems on low dimensional
CW -complexes. This `occasional' relation can be seen in combinatorial group th*
*eory
that both combinatorial descriptions ask the same combinatorial question how to
understand the quotient groups (R \ S)=[R; S] for certain subgroups R and S of a
finitely generated free group. In Theorem 1.4, the subgroups R and S are partic*
*ularly
given.
These combinatorial methods extend in a second direction which we now describ*
*e.
It is well known that any space is weakly homotopy equivalent to a minimal simp*
*licial
set. An important feature of minimal simplicial sets is that these are the sma*
*llest
(fibrant) simplicial sets with which we can still do homotopy theory. There hav*
*e been
uses of computers for studying homotopy groups. Minimal simplicial sets become
very important in computer homotopy theory. A minimal simplicial group means
a minimal simplicial set that is also a simplicial group. The group structure *
*in a
minimal simplicial group helps us to control the homotopy groups. Moore's probl*
*em
is whether a loop space has the (weak) homotopy type of a minimal simplicial gr*
*oup.
In [1], Adams proved that if X is a two stage Postnikov system X with the only
possible non-trivial homotopy groups being ssn(X) and ssn+1(X), then X has the
homotopy type of a minimal simplicial group. In [28], we proved that a two sta*
*ge
Postnikov system X with the only non-trivial homotopy groups being ssn(X) = Z=2
and ssn+i(X) = Z=2 with n; i > 0 has the homotopy type of a minimal simplicial
group if and only if the Postnikov invariant is trivial or Sqi+1. Milnor proved*
* that the
three stage Postnikov system obtained by taking the first three non-trivial hom*
*otopy
groups of Sn with n > 1 does not have the homotopy type of a minimal simplicial
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 5
group. Unfortunately , this important example was never published. We will use
Cohen's K-construction on pointed simplicial sets [7] to reprove Milnor's examp*
*le in
this paper. (See Proposition 6.19.)
Cohen's K-construction arose in studying self-maps of the loop space of a sus-
pension. This construction is an important tool in studying the exponent probl*
*em
in homotopy theory. Also important relations between Cohen's construction and
Goodwillie's tower of the identity functor were found by Bill Dwyer and other p*
*eo-
ple recently. Cohen [7] showed that the homotopy classes of all self maps that*
* are
natural in X of X form a progroup which is built up from the Moore complex of
Milnor's construction F (S1). Thus Theorem 4.3 provides detailed information ab*
*out
this progroup. On the other hand, the homotopy classes of all functorial self m*
*aps
of the loop spaces of a double suspension form a progroup which is built up by *
*the
Moore complex of Cohen's construction K(S1) instead of F (S1) [7]. By using The-
orem 1.2, we determine the general homotopy groups of K(S1). (See Theorem 6.7.)
Furthermore, the Lie structure of ss*(K(S1)) is given. (See Proposition 6.10.) *
*We find
out that in addition to considerable applications of the homotopy groups of K(S*
*1)
to Mathematical Physics, the simplicial group K(S1) itself plays an important r*
*ole
classifying certain type of minimal simplicial groups. ( See Theorem 6.14.). *
*Theo-
rems 6.7 and 6.14 are used to show that the three stage Postnikov tower is not *
*homo-
topy equivalent to a minimal simplicial group. In addition, Cohen's K-construct*
*ion
is extensively used to study functorial self coalgebra maps of primitively gene*
*rated
tensor algebras with applications to functorial decompositions of loop suspensi*
*ons.
(See [25].) It is known (unpublished) that all functorial self coalgebra maps o*
*f tensor
algebras (without assuming that the Hopf structure is primitively generated) ov*
*er
the prime field Fp form a progroup which is built up from the p-completion grou*
*ps
of the Moore complex of F (S1). This raises possibilities to study the homotopy*
* by
using representations of the Moore complex into the convolution algebra of self*
* linear
maps of tensor algebras.
The article is organized as follows. In Section 2, we study some general prop*
*erties
of simplicial groups. Central extensions in the Moore-Postnikov systems will be
studied. In Section 3, we study the intersection of certain subgroups in free g*
*roups.
The proofs of Theorems 1.2 and 1.4 is given in Section 4, while Theorem 4.5 is
Theorem 1.2 and Theorem 4.13 is Theorem 1.4. The proof of Theorem 1.5 is given *
*in
Section 5, while Theorem 5.10 is Theorem 1.5. In Section 6, we give some applic*
*ations
of our descriptions. One example is to compute the homotopy groups of Cohen's
construction on the 1-sphere.
The author is indebted to helpful discussions with J. C. Moore and F. R. Cohen
and would like to thank D. Kan, M. Mahowald, J. P. Serre, J. Stasheff and many
other mathematicians for their kindly encouragement and helpful suggestions.
6 JIE WU
2. Central Extensions in simplicial group theory
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, where k[n] is the subcomplex of [n] generated by all di(oen)
for i 6= k and oen is the nondegenerate n simplex in [n], and where dj is one of
the face functions [14, 9]. Recall that any simplicial group is a Kan complex *
*[19].
GivenTa simplicial group G, the Moore chain complex (NG; d0) is defined by NGn =
j6=0Ker(dj) together with d0: NGn ! NGn-1. The classical Moore Theorem is that
ssn(G) ~= Hn(NG) (seeT[19, Theorem 4], [9, Theorem 3.7] or [15, Proposition 5.4*
*]].
Now let Zn = Zn(G) = jKer(dj) denote the cycles and let Bn = BGn = d0(NGn+1)
denote the boundaries. It is easy to check that Bn is a normal subgroup of Gn *
*for
any simplicial group G.
Let G be a simplicial group and let ff 2 G0. The simplicial automorphism OEff*
*:G !
G is defined by OEff(x) = (sn0ff)-1xsn0ff for x 2 Gn, where sj is a degeneracy *
*function.
Suppose that ff is homotopic to fi,that is, there exists an element fl 2 G1 suc*
*h that
d0fl = ff and d1fl = fi. Then it is easy to check that OEffis homotopic to OEf*
*i. Thus
ss0(G) acts on ss*(G). Notice that ss0(G) acts on itself by conjugation.
Lemma 2.1. Let G be a simplicial group and let n 0. Suppose that ss0(G) acts
trivially on ssn(G). Then the homotopy group ssn(G) is contained in the center*
* of
Gn=BGn.
Proof.If n = 0, then ss0(G) ~= G0=BG0, which is abelian if ss0(G) acts triviall*
*y on
itself. Thus we may assume that n 1. Notice that ssn(G) ~=ZGn=BGn. It suffices
to show that the commutator [x; y] 2 BGn for any x 2 ZGn and y 2 Gn.
Now let x 2 ZGn and let y 2 Gn. Let z denote sn0dn0(y-1) . y. 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. Let the simplic*
*ial
map fz: [n] ! G be the representative of z, i.e, fz(on) = z for the nondegenera*
*te
n-simplex on. Consider the simplicial map
[fx; fz]: Sn x [n] ! G
defined by [fx; fz](a; b) = [fx(a); fz(b)] = (fx(a))-1(fz(b))-1fx(a)fz(b), the *
*commuta-
tor of fx(a) and fz(b). Notice that
fz(dn0(o)) = dn0sn0dn0(y-1) . dn0y = 1:
Let v be the subsimplicial set of [n] generated by the vertex dn0o and let * be*
* the
base-point of Sn. Then [fx; fz] is trivial restricted to the subsimplicial set *
*Sn _[n] =
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 7
Sn x v [ * x [n]. Thus [fx; fz] factors through the quotient simplicial set Sn *
*^ [n].
Let OE be the composite
j n [fx; fz]
Sn ______-S ^ [n] ________-G;
where j(oen) = oen ^ on. Notice that OE(oen) = [x; z]. Let {[x; z]} denote th*
*e homo-
topy class in ssn(G) represented by the cycle [x; z]. Then the map OE: Sn ! G *
*is
a representative of the cycle [x; z] and OE*(n) = {[x; z]}, where the homomorph*
*ism
OE*: ss*(Sn) ! ss*(G) is induced by the map OE, where n is a generator for ssn(*
*Sn).
Notice that Sn ^ [n] is contractible. Thus {[x; z]} = 0 in ssn(G) and so [x; z]*
* 2 BGn.
Let ff = dn0y. Then {(sn0ff)-1xsn0ff} = {x} since ss0(G) acts trivially on s*
*sn(G).
Thus [x; sn0ff] 2 BGn and so [x; y] = [x; sn0ffz] 2 BGn by the Witt-Hall identi*
*ty that __
[a; bc] = [a; c][a; b][[a; b]; c]: The assertion follows. *
* |__|
Remark 2.2. Let G be a simplicial group. Then ss0(G) acts trivially on ssn(G)*
* if
and only if the classifying space BG is n + 1-simple.
Corollary 2.3. Let G be a simplicial group. Suppose that BG is simple. Then the
homotopy group ssn(G) is contained in the center of Gn=BGn for each n.
Corollary 2.4. Let G be a connected simplicial group. Then the homotopy group
ssn(G) is contained in the center of Gn=BGn for each n.
Now we consider the Moore-Postnikov systems of simplicial groups.
Definition 2.5. Let G be a simplicial group. The subsimplicial group RnG is def*
*ined
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 groups of*
* G.
Thus_both PnG and P nG are quotient simplicial groups of G. Notice that RnG
RnG Rn-1G: There is a cofiltration
__ __
G ! . .!.PnG ! PnG ! Pn-1G ! . .!.P0G:
By checking the definition of Moore-Postnikov systems of a simplicial set [9, 1*
*9, 20,
21], we have
Lemma 2.6. The quotient simplicial group PnG is the standard n-th stage of the
Moore-Postnikov system of the simplicial group G.
8 JIE WU
__
The quotient simplicial group P nG has the same homotopy type as PnG.
__
Lemma 2.7. The quotient simplicial homomorphism qn: PnG ! PnG 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
: NG
q for q < n + 1.
__
The Moore chain complex of P nG is as follows:
8
__ < 1 for q > n
N(P nG)q = NGn=BGn for q = n
: NG
q for q < n.
__ *
* __
Thus (qn)*: ss*(PnG) ! ss*(P nG) is an isomorphism and the assertion follows.*
* |__|
__ __
Let F n denote the kernel of the quotient simplicial homomorphism rn: PnG !
Pn-1G for n > 0.
__
Lemma 2.8. The simplicial group F nis the minimal simplicial group K(ssnG; n)*
* for
each n > 0.
__
Proof.The Moore chain complex of F nG is as follows:
ae
__ 1 for q 6= n
N(F nG)q = ss
nG for q = n.
*
*__
The assertion follows. |*
*__|
Lemma 2.9. Let G be a simplicial group. Suppose that ss0(G) acts trivially on*
* ssn(G).
Then the short exact sequence of simplicial groups
__ __
0 ! FnG ! PnG ! Pn-1G ! 0
is a central extension.
__ __
Proof.Consider the relative commutator subsimplicial group [F nG; PnG]. Notice
that __
F nGn = ZGn=BGn ~=ssn(G)
and __
PnGn = Gn=BGn:
__ __ __ __
By Lemma 2.1, [F nG; PnG]n = 1. Notice_that [F nG; PnG] is_a subsimplicial_group
of the minimal simplicial group F nG ~=K(ssnG;_n)._Thus [F nG; PnG] is a minimal
*
* __
simplicial group of type K(ss; n) and so [F nG; PnG] = 1. The assertion follows*
*. |__|
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 9
Corollary 2.10. Let G be a simplicial group. Suppose that BG is simple. Then the
short exact sequence of simplicial groups
__ __
0 ! FnG ! PnG ! Pn-1G ! 0
is a central extension for each n.
Corollary 2.11. Let G be a connected simplicial group. Then the short exact se-
quence of simplicial groups
__ __
0 ! FnG ! PnG ! Pn-1G ! 0
is a central extension for each n.
By Lemmas 2.7, 2.8 and 2.9, we have
Theorem 2.12. Let G be a simplicial group. Suppose that BG is simple. Then th*
*ere
is a cofiltration
__ __
G ! . .!.PnG ! PnG ! Pn-1G ! . .!.P0G:
of G such that:
__
1) the quotient simplicial homomorphism qn: PnG ! P nG is a homotopy equiv-
alence for each_n; __
2) the kernel F nG of the quotient simplicial homomorphism rn: PnG ! Pn-1G
is the minimal simplicial group K(ssn(G); n) for each n;
3) the short exact sequence of simplicial groups
__ __
0 ! FnG ! PnG ! Pn-1G ! 0
is a central extension for each n;
4) G is the inverse limit of the cofiltration;
5) G is the homotopy inverse limit of the cofiltration.
Corollary 2.13. Let G be a connected minimal simplicial group and let {PnG} be
the Moore-Postnikov systems of G. Then the short exact sequence of simplicial g*
*roups
0 ! FnG ! PnG ! Pn-1G ! 0
is a central extension for each n.
In some cases, the homotopy group ssn(G) is the same as the center of Gn=BGn.
Definition 2.14. A simplicial group is said to be r-centerless if the center Z(*
*Gn) =
{1} for n r.
Proposition 2.15. Let G be a reduced r-centerless simplicial group. Then ssn(G*
*) ~=
Z(Gn=BGn) for n r + 1
10 JIE WU
Proof.By Lemma 2.1, ZGn=BGn Z(Gn=BGn). It suffices to show that Z(Gn=BGn)
ZGn=BGn for n r + 1. Now let "x2 Z(Gn=BGn). Choose x 2 Gn with p(x) = "x,
where p: Gn ! Gn=BGn is the quotient homomorphism. To check "x2 ZGn=BGn,
it suffices to show that x 2 ZGn 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-1y] for j*
* > 0 and
[d0x; y] = d0[x; s0y]. Since "x2 Z(Gn=BGn), [x; z] 2 BGn ZGn for all z 2 Gn an*
*d __
therefore [djx; y] = 1 for all y 2 Gn-1. The assertion follows. *
* |__|
By inspecting the proof, we also have
Proposition 2.16. Let G be a reduced r-centerless simplicial group. Then Z(Gn=*
*ZGn) =
{1} for n r + 1.
Lemma 2.17. Let G be a reduced simplicial group so that Gn is cyclic or cente*
*rless
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 dq0Osq*
*0: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 2.18. Let G be a reduced simplicial group such that Gn is cyclic or c*
*en-
terless for each n. Then ssn(G) ~=Z(Gn=BGn) for n 6= flG + 1, where flG is defi*
*ned
as above.
Notice that, for any free group F , rank(F ) 2 , Z(F ) = {1} and F 6= {1}. We
have
Proposition 2.19. Let G be a reduced simplicial group such that Gn is a free g*
*roup
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 . Furthermore, ssn(G) ~=Z(Gn=BGn) for n 6= flG + 1.
Let G be a group and let nG be the descending central series of G starting wi*
*th
1 = G. Let Tor(G) = {x 2 G|xq = 1for some q > 0}. If G is an abelian group,
then Tor(G) is the torsion subgroup of G. For general group G, Tor(G) may only
be a subset of G. Let G be a connected free simplicial group. Then the torsion
component of the homotopy groups ss*(G) can be described as followings. In the
following theorem, a free simplicial group is defined in [9].
Theorem 2.20. Let G be a connected free simplicial group such that ssj(G) = 0*
* for
j q with some q 0 and let s 2n-q. Then Tor(Gn=) is a subgroup of
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 11
Gn=. Furthermore, there are isomorphisms of groups
Tor(ssn(G)) ~=Tor(Z(Gn=)) ~=Tor(Gn=);
where s = sGn.
Proof.Notice that sG = {sGn}n0 is a normal subsimplicial group of G. Let
q :G ! G=sG be the quotient simplicial homomorphism. Then N(q): NGn !
N(G=sG)n is an epimorphism for each n 0 and so B(q): BGn ! B(G=sG)n is
an epimorphism for each n 0. Thus there is a canonical isomorphism of groups
(G=sG)n____~ s
= Gn=:
B(G=sG)n
By the Curtis Theorem [8], we have ssj(sG) = 0 for j n and so q*: ssj(G) !
ssj(G=sG) is an isomorphism for j n. By Lemma 2.1, the composite
q*Z(G=sG)n s
ssn(G) = ZGn=BGn ___~-_________ __-Z(Gn=)
= B(G=sG)n
is a monomorphism. Thus the composite
Tor(ssn(G))___-Tor(Z(Gn=)) __-Tor(Gn=)
is one to one. It suffices to show that this composite is an epimorphism.
Let x 2 Tor(Gn=) and let ff 2 (G=sG)n such that x = ffB(G=sG)n.
Then there exist k > 0 such that ffk 2 B(G=sG)n Z(G=sG)n. Thus, for each
0 j n, we have
(dj(ff))k = dj(ffk) = 1
in the group (G=sG)n-1 = Gn-1=sGn-1. Notice that (G=sG)n-1 = Gn-1=sGn-1
is a torsion-free group. Thus
dj(ff) = 1
for each 0 j n and so ff 2 ZGn. Hence
Z(G=sG)n s
x 2 ___________\ Tor(Gn=)
B(G=sG)n
*
*__
and the assertion follows. |*
*__|
Remark 2.21. Let X be a simply connected reduced simplicial set and let GX be*
* the
Kan-construction for X. Then GX is a free simplicial group in the sense of [9].*
* Thus
this theorem shows that the torsion component of any homotopy group of a simply
connected space is the torsion component of the center of certain nilpotent gro*
*up.
Example 2.22. The 1-stem is determined in this example.
12 JIE WU
Let G = F (Sn), Milnor's F -construction on the standard n-sphere for n 1. T*
*hen
Gn ~=F (oe) ~=Z(oe), the free abelian group generated by oe, Gn+1 ~=F (s0oe; s1*
*oe; : :;: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 = [ZGn+1; Gn+1] BGn+1:
If n = 1, then it will be shown that 3Gn+1 = BGn+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
ae
1 for k 6= j
dk([sj-1sioe; sj+1sjoe]) = [s
ioe; sjoe]fork = j
for i + 1 < j n,
8
< [si+1oe; si+2oe]fork = i + 1
dk[si+2si+1oe; si+3sioe] = [si+1oe; sioe]for k = i + 3
: 1 otherwise
and 8
>> [si+1oe; si+2oe]fork = i + 1
< [s oe; s oe]for k = i + 2
dk[si+2sioe; si+3si+1oe] = i i+2
>> [sioe; si+1oe]for k = i + 3
: 1 otherwise
By the Homotopy Addition Theorem [9, 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). T*
*hus
[s0oe; s1oe] =2BGn+1 and, by the relations above,
ssn+2(Sn+1) ~=ssn+1(G) ~=ZGn+1=BGn+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 study the intersections of certain subgroups in free grou*
*ps.
Definition 3.1. let S be a set and let T S a subset. The projection homomor-
phism
ss :F (S) ! F (T )
is defined by ae
x if x 2 T
ss(x) = 1 if x 62 T
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 13
Now let ss :F (S) ! F (T ) be the projection homomorphism and let R equal the
kernel of ss. Define subsets of the free group F (S) as follows.
AT(k) = {[[x; yffl11]; . .].; yffltt]|0 t k; fflj = 1; y = yffl11:y:f:flt*
*t2 F (T ); x 2 S - T };
where y = yffl11.y.f.fltt2 F (T ) runs over reduced words in F (T ) with t k a*
*nd yj 2 T .
Furthermore define [[x; yffl11]; : :]:; yffltt] = x for t = 0. Let
BT(k) = {OE-1xOE|OE 2 F (T ) a reduced word with length; l(OE) k; x 2 S - T*
* };
AT = [k0 AT(k)
and
BT = [k0 BT(k):
By the classical Kurosch-Schreier theorem ( see [17, 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 abelianlization of the group*
* F .
Then OE: F1 ! F2 is a monomorphism.
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 series of the group G starting w*
*ith
1G = G and so
lim F1=rF1 ! limF2=rF2
r r
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. Let ik: AT(k) ! R and
jk: BT(k) ! R be 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; . .;.y*
*ffltt] 2
AT(k). If t < k, then
w 2 AT(k - 1) F (BT(k - 1)) F (BT(k));
14 JIE WU
by induction. Now
[[x; yffl11]; . .].; yfflkk] = [[x; yffl11]; . .].; yfflk-1k-1]-1 . y-*
*fflkk[[x; yffl11]; . .].; yfflk-1k-1]yfflkk
Notice that [[x; yffl11]; . .].; yfflk-1k-1] 2 F (BT(k - 1)) by induction. Then
Ys
[[x; yffl11]; . .].; yfflk-1k-1] = (OE-1jxjOEj))jj
j=1
for some 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 homomorphism "ik
is 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 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-1xOE 2 Im 'k by induction. Let OE = yffl1*
*1.y.f.flkk
be 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; yfflkk]. Notice that z 2 Im ("ik) by induction. It suffice*
*s 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 t*
*his 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, th*
*en [w; yffl] 2
F (AT(k)) by definition. Suppose that yffl11.y.f.flttyfflis not a reduced wor*
*d. 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 finished and so the first indu*
*ction
is finished. The assertion follows.
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 15
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.
Now consider the general case. By Proposition 3.2, 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 S0and T 0of S and T , respectively, so that *
*w 2 BT0.
By the special case as above, "i|F(AT0):F (AT0) ! F (BT0) is an isomorphism and
w 2 Im ("i|F(AT0)). Thus "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 homomorphisms. *
*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 Definition 3.1. Let
T2(2)= {w 2 A(T1)|w = [[x; yffl11]; . .;.yffltt] with x; yj 2 T2 for al*
*l}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 elements*
* 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;fo*
*r all j
where [xffl11; . .;.xfflll] are the elements in A(T1; T2; . .;.Tk-1) which are *
*written down
as commutators. Then let A(T1; T2; . .;.Tk) be defined by
A(T1; T2; . .;.Tk) = A(T1; T2; . .;.Tk-1)T(k):
k
16 JIE WU
Theorem 3.5. Let S be a set and let Tj be a subset of S for 1 j k. Let
ssj: F (S)T! F (Tj) be the projection homomorphism for 1 j k. Then the inter-
section kj=1Ker(ssj) is a free group freely generated by A(T1; T2; . .;.Tk).
Proof.The proof is given by inductionTon k. If k = 1, the assertion follows fr*
*om
the above lemma. Suppose that k-1j=1Ker(ssj) = F (A(T1; T2; . .;.Tk-1)) and *
*con-
T k
sider ssk: F (S) ! F (Tk). Then j=1Ker(ssj) = Ker(ssk:F (A(T1; T2; . .;.Tk-1*
*)) !
F (Tk)), where sskis ssk restricted to the subgroup F (A(T1; T2; . .;.Tk-1)). L*
*et 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 proj*
*ection ho-
momorphism. 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)
6| 6| 6||
| | |
| |j |
| | |
| | |
| | |
= (k)
F (Tk(k))________-F (Tk )___-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
Ker (ssk) = Ker(F (A(T1; T2; . .;.Tk-1)) ! F (Tk(k)) = F (A(T1; T2; . .;.Tk*
*)):
*
*__
The assertion follows. |*
*__|
Corollary 3.6. Let ssj be theTprojection homomorphisms as in Theorem 3.5. Then
the intersection subgroup kj=1Ker(ssj) equals the commutator subgroup [[;*
* . .;.]]
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 Milnor's construction 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 4.13 is Theorem 1.4. Recall that the simplicial 1-sphere
S1 is a free simplicial set generated by a 1-simplex oe. Thus S10= {*}, S11= {o*
*e; *}
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.
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 17
Let yi denote xix-1i+1for -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 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.
Now let Cn+1 be the subgroup of F (y0; . .;.yn) generated by all of the commu*
*tators
[yffl1i1; . .;.yffltit] where
1) fflj = 1;
2) 0 is n;
3) all integers in {0; 1; . .;.n} appear as at least one of the integers is;
4) for each t n + 1, [. .].runs over all of the commutator bracket arrangem*
*ents
of weight t.
Lemma 4.2. The group Cn+1 is a subgroup of NF (S1)n+1. That is
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.Let S = {y0; y1; . .;.yn} and let Tj = {y0; . .;.^yj. .;.yn} for 0 j n.*
* 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);
18 JIE WU
where ae
dj(yk) = yk for 0 k j - 2;
yk-1 for j k n:
Thus Ker(dj) = Ker(ssj) and so
n+1" n+1"
NF (S1)n+1 = Ker(dj) = Ker(ssj):
j=1 j=1
By Theorem 3.5, we have
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 fo*
*llowing
statement.
Statement: Given each 0 j n and let
w = [yffl1i1; yffl2i2; . .;.yffltit] 2 A(T0; T1; . .;.Tj):
Then the set {y0; y1; . .;.yj} is a subset of the set {yi1; yi2; . .;.yit}.
We show this statement by induction on j. Notice that
F (T0) = F (y1; . .;.yn):
For j = 0, we have
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}
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 19
for each w = [yffl1i1; . .;.yffltit] 2 A(T0; T1; . .;.Tj). The induction is fi*
*nished_and the
theorem follows. |_*
*_|
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; yffl2i2; . .;.yffltit]
where
1) ffl = 1;
2) -1 is n with y-1 = (y0y1. .y.n)-1;
3) all elements in the set {y-1; y0; . .;.yn} appear as at least one of the *
*elements
yis;
4) for each t n + 2, [. .].runs over all bracket arrangements of weight 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; . .;.yffltit]
with {yi1; yi2; . .;.yit} = {y0; y1; . .;.yn} as sets, where fflj = 1. By Propo*
*sition 2.15,
it suffices to check that
ss2(F (S1)) ~=Z(F (S1)2=B2) = Z(F (y0; y1)=B2):
By Example 2.22, we have
Z2 = 2F (S1)2 = 2F (y0; y1)
and so 3F (y0; y1) B2. Now , by the construction of B2, we have B2 3F (y0; y1*
*).
Thus
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, that is, many of t*
*hem
can be cancelled out.
Let C0n+1 be the subgroup of F (S1)n+1 generated by all commutators of the fo*
*rm
[[yffl1i1; yffl2i2]; . .;.yffltit]; where
1) ffl = 1;
20 JIE WU
2) 0 is n;
3) all elements in {y0; y1; . .;.yn} appear as one of the elements yis.
Proposition 4.7. There is an isomorphism of groups
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*
*+1
and wx1x2. .x.s2 s+1 for each s. In fact, if this statement holds, then
wx1x2. .x.s-1 1 mod s \ NF (S1)n+1
for each s and
w = (wx1. .x.s-1) . (x1x2. .x.s-1)-1 2 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. Notice that C0n+1
NF (S1)n+1. Thus
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 homomorphism. Then the ele-
ment ss(wx1. .x.s-1) is a linear combination of basic Lie products. We claim t*
*hat
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 do*
*es not
appear and c is a linear combination of basic Lie products in which yj appears.*
* Now
the face homomorphism dj+1:F (y0; . .;.yn) ! F (y0; . .;.yn-1) induces a homomo*
*r-
phism of abelian groups
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)
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 21
is an isomorphism. Thus b = 1. This contradicts to that b is a nontrivial lin*
*ear
combination of a basis. Thus ss(wx1. .x.s-1) is a linear combination of basic *
*Lie
products in which all of the yj appear. By [17, Theorem 5.12], there exists xs *
*in C0n+1
so that ss(wx1. .x.s) = 1, or wx1. .x.s2 s+1. The induction is finished now an*
*d_
the assertion follows. |*
*__|
Let B0n+1 be the subgroup of F (S1)n+1 generated by all commutators of the fo*
*rm
[[yffl1i1; yffl2i2]; . .;.yffltit]; where
1) ffl = 1;
2) -1 is n with y-1 = y0y1. .y.n;
3) all elements in {y-1; y0; y1; . .;.yn} appear as one of the elements yis.
Corollary 4.8. There is an isomorphism of groups
__BF_(S1)n+1___~ B0n+1
= _________0
s \ BF (S1)n+1 s \ Bn+1
for each s.
Let Dsn+1be the subgroup of F (S1)n+1 = F (y0; y1; . .;.yn) generated by all *
*com-
mutators of the form [[yffl1i1; yffl2i2]; . .;.yffltit]; where
1) ffl = 1;
2) -1 is n with y-1 = y0y1. .y.n;
3) either all elements in {y-1; y0; y1; . .;.yn} appear as one of the elemen*
*ts yis
or t s.
Notice that ssn(S2) is a finite group for n 4. By Theorem 2.20, we have
n+1 1 *
*2n+1
Proposition 4.9. Let n 2. Then Tor(F (S1)n+1=D2n+1) is subgroup of F (S )n+1=D*
*n+1 .
Furthermore there are isomorphisms of groups
F (S1)n+1 F (S1)n+1
ssn+2(S2) ~=Tor Z _________n+1~= Tor _________n+1:
D2n+1 D2n+1
n+1
Remark 4.10. The group F (S1)n+1=D2n+1 is a finitely presented nilpotent group
with explicit generators and relations.
By Corollary 3.6, we have
Theorem 4.11. In the free group F (y0; . .;.yn), we have the identifications *
*of sub-
groups
NF (S1)n+1 = [[; . .;.]];
BF (S1)n+1 = [[; ; . .;.]]:
Thus Theorem 4.5 can be rewritten as follows.
22 JIE WU
Theorem 4.12. In the free group F (y0; . .;.yn) for n 1, the center
Z(F (y0; . .;.yn)=[[; ; . .;.]]) ~=ssn+2(S3)
By Lemma 4.1 and Theorem 3.5, Ker(d0) = = and therefore
Zn+1 = [[; . .;.]] \ :
Thus we have
Theorem 4.13 (Theorem 1.4). In the free group F (y0; . .;.yn) with n 1,
[[;_._.;.]]_\_~ 3
= ssn+2(S ):
[[; ; . .;.]]
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 Theorem 5.10*
* 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). Recall*
* that
(_ff2JS1)0 = * and (_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)jdenote x(ff)j.(x(ff)j+1)-1 for -1 j n, where xff-1= xffn+1= 1 in F *
*(_ff2JS1)n+1 =
F (x(ff)0; x(ff)1; . .;.x(ff)n|ff 2 J): 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 (ff)
< yk-1 for j k;
dj(y(ff)k) = 1 for j = k + 1;
: (ff)
yk for j > k + 1,
and 8
>< y(ff)k+1 for j k;
sj(y(ff)k) = y(ff). y(ff)for j = k + 1;
>: k (ff) k+1
yk for 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 commut*
*ators
of the form
[y(ff1)ffl1i1; . .;.y(fft)ffltit];
where
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 23
1) ffl = 1;
2) 0 is n;
3) ffj 2 J;
4) all integers in {0; 1; . .;.n} appear as at least one of the integers is;
5) for each t n + 1, [. .].runs over all of the commutator bracket arrangem*
*ents
of weight t.
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,
Ker(dj) \ F (_ff2JS1)n+1 = ;
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
| |
?| ?|
= (ff)
F (y(ff)j|0 j n - 1; ff 2_J)_-F (yj |0 j n - 1; ff 2 J);
where p is the projection and
(
(ff)
djy(ff)k= yk-1 for j k;
y(ff)kfor j > 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 = Ker(dj) = F (A(T0; T1; . .;.Tn))
j=1
24 JIE WU
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. |_*
*_|
Lemma 5.5. For 0 j n, let W = [y(ff1)ffl1i1; . .;.y(fft)ffltit] 2 A(T0; T1;*
* . .;.Tj). Then
the set {0; 1; . .;.j} is contained in the set {i1; . .;.it}.
Proof.The proof is given by induction on j for 0 j n. Notice that, by the
construction, each element in A(T0; T1; . .;.Tj) is written as a certain commut*
*ator.
If j = 0, then
A(T0) = {y(ff)0; [[y(ff)0; y(ff1)ffl1i1]; . .;.]; y(fft)fflt*
*it]}
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;w.i.;.y(fft)*
*ffltit]thy(ffs)is2}Tj:
Notice that y(ffs)is2 Tj () is 6= j. Let W = [y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.*
*y(fft)ffltit] 2 A(T0; . .;.Tj-1)-
Tj(j). Then there exists some is with 1 s t such that is = j. By induction, t*
*he set
{0; . .;.j -1} is contained in the set {i1; . .;.it}. Thus {0; 1; . .;.j} {i1;*
* i2; . .;.it}.
Recall that, by construction,
A(T0; . .;.Tj) = AT(j):
j
Thus we have
{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.
|__|
Corollary 5.6. The Moore chain complex
NF (_ff2JS1)n+1 n+1F (_ff2JS1)n+1
for n 0, where qG is the q-th term in the lower central series of a group G st*
*arting
with 1G = G.
Theorem 5.7. ssn+2(_ff2JS2) is isomorphic to the center of the group with gen*
*erators
y(ff)0; y(ff)1; . .;.y(ff)n
for ff 2 J. and relations of the form
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit];
where
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 25
1) ffl = 1;
2) -1 is n with yff-1= (yff0. yff1. .y.ffn)-1;
3) ffj 2 J;
4) all integers in {-1; 0; 1; . .;.n} appear as at least one of the integers*
* is;
5) for each t n + 2, [. .].runs over all of the commutator bracket arrangem*
*ents
of weight t.
Proof.Notice that ssn+2(_ff2JS2) ~=ssn+1(F (_ff2JS1)). By the above theorem, Bn*
*+1
is generated by
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit]
By Proposition 2.15, 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 ~=ff2JZ ~=ss2(_ff2JS2):
*
* __
The assertion holds for all of n. *
* |__|
For the general case, we need a simplicial group construction.
Definition 5.8. Let G be a simplicial group and let X be a pointed simplicial s*
*et
with a base-point *. The simplicial group FG (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 *
*face
and degeneracy homomorphisms in F G(X) are given in the canonical way by the un*
*i-
versal property of the coproduct in the category of groups and group homomorphi*
*sms.
Lemma 5.9 ( [6], Theorem 9, pp.88). Let G be a simplicial group and let X be a
pointed simplicial set. Then the geometric realization |F G(X)| is homotopy equ*
*ivalent
to (B|G| ^ |X|).
A generalization of this lemma by using fibrewise simplicial groups is given *
*in [27].
Theorem 5.10 (Theorem 1.5). Let ss be any group and let {x(ff)|ff 2 J} be a s*
*et of
generators for ss. Let (ss)j be a copy of ss with generators {x(ff)j|ff 2 J}. *
* Then, for
n 6= 1, ssn+2(K(ss; 1)) is isomorphic to the center of the quotient group of th*
*e free
product
groupsa
(ss)j
0jn
modulo the relations of the form
[y(ff1)ffl1i1; y(ff2)ffl2i2; . .;.y(fft)ffltit];
where
26 JIE WU
1) ffl = 1;
2) -1 is n with yff-1= (yff0. yff1. .y.ffn)-1;
3) ffj 2 J;
4) all integers in {-1; 0; 1; . .;.n} appear as at least one of the integers*
* is;
5) for each t n + 2, [. .].runs over all of the commutator bracket arrangem*
*ents
of weight t.
Proof.Since {x(ff)|ff 2 J} is a set of generators for ss, F (x(ff)|ff 2 J) ! ss*
* 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 5.7, Lemma 5.9 and Propo-__
sition 2.15. |_*
*_|
Example 5.11. Let ss be an abelian group. Then
ss3(K(ss; 1)) ~=2(ss * ss)=3(ss * ss) ~=ss ss;
where G * H is the free product of the groups G and H.
Proof.Consider the simplicial group F ss(S1). Then F ss(S1)1 = ss and F ss(S1)2*
* = ss*ss.
Notice that ss is abelian. Then the commutator subgroup 2(ss * ss) is contained*
* in
the cycles ZF ss(S1)2. By Lemma 2.1, the subgroup 3(ss * ss) is contained in t*
*he
boundaries BF ss(S1)2. By Corollary 5.6 and Theorem 5.10, there are equations
ZF ss(S1)2 NF ss(S1)2 2(ss * ss)
and
BF ss(S1)2 3(ss * ss):
Thus
ZF ss(S1)2 = 2(ss * ss)
and
BF ss(S1)2 = 3(ss * ss):
*
*__
The assertion follows. |*
*__|
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 27
6.Applications
In this section, we study Cohen's K-construction. Our descriptions for the ho*
*mo-
topy groups of the 3-sphere allows us to give a calculation for the K-construct*
*ion of
the 1-sphere.
Definition 6.1. Let X be set. The group K(X) is defined to the the quotient gr*
*oup of
the free group F (X) modulo the normal subgroup generated by all of the commuta*
*tors
[[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 q*
*uotient
simplicial group of F (S) modulo the normal simplicial subgroup generated by al*
*l of
the commutators [[x1; x2]; . .;.]; xt] with xi2 S and xi= xj for some 1 i < j *
* t,
where F (S) is Milnor's F (K)-construction for the simplicial set S.
Definition 6.2. The group Lie(n) consists of the elements of weight n in the Lie
algebra Lie(x1; x2; . .;.xn) which is the quotient Lie algebra of the free Lie *
*algebra
L(x1; x2; . .;.xn) over Z modulo the two sided Lie ideal generated by the Lie e*
*lements
[[xi1; xi2]; . .;.]; xit] with il= ik for some 1 l < k t.
The following lemmas are due to Fred Cohen.
Lemma 6.3 ( [7]). 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 ( [7]). In the group K(x1; x2; . .;.xn), the normal subgroup generated
by xj is abelian for each 1 j n.
Lemma 6.5 ( [7]). The set {[[x1; xoe(2)]; . .;.xoe(n)]|oe 2 n-1} is a Z-basis f*
*or 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 degenerate functions si:S1*
*n+1!
S1n+2are as follows: ae
xj for j < i,
dixj = x
j-1 for j i,
and ae
xj for j < i,
sixj = x
j+1 for j i;
where we put x-1 = * and xn = * in S1n.
28 JIE WU
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 c*
*om-
mutators
[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:
Notice that K(S1)n+1 ~= K(x0; x1; . .;.xn). Thus n+1K(S1)n+1 ~= Lie(n + 1) a*
*nd
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
NK(S1)n+1 = n+1K(S1)n+1 ~=Lie(n + 1)
*
* __
with d0: NK(S1)n+1 ! NK(S1)n is a trivial differential. The assertion follows*
*. |__|
Remark 6.8. Let n act on Lie(n) by permuting letters. The R(n)-module LieR(n*
*) =
Lie(n) R has important applications in Representation Theory and Mathematical
Physics.
Corollary 6.9. Let ss :F (S1) ! K(S1) be the quotient simplicial homomorphism.
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 . .^.s*
*j.s.0.oe
in S1n+1. The following lemma follows directly from Lemma 6.6.
Lemma 6.10. Let I = (i1; i2; . .;.im ) be a sequence with i1 < i2 < . .<.im*
* . Then
sI: S1n+1- * ! S1n+m+1- * is the composite
sI
{x0; x1; . .;.xn}_-{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 [8] is d*
*efined 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*
* all shuffles
of (0; 1; . .;.m + n - 1), that is, all permutations, so that a1 < a2 < . . .*
*< an,
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 29
b1 < b2
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} an*
*d 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. |*
*__|
L 1
Let LieR = n=1 Lie(n) R be a graded module over a ring R with the graded
paring defined in Proposition 6.11. Then LieR is a grade Lie algebra.
Corollary 6.12. There is an isomorphism of Hopf algebras
H*(K(S1); Q) ~=U(LieQ);
where U(L) is the universal enveloping algebra of a Lie algebra.
Definition 6.13. A simplicial group is minimal if it is also a minimal simplici*
*al
set.
Recall that a simplicial group G is minimal if and only if the Moore chain co*
*mplex
NG is minimal [9].
30 JIE WU
Theorem 6.14. The simplicial group K(S1) is the universal minimal simplicial *
*quo-
tient 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 homomorphism
f :F (S1) ! G factors through K(S1).
Proof.By inspecting the proof of Theorem 6.7, K(S1) is a minimal simplicial gro*
*up.
The assertion (2) follows from the following statement.
Statement: K(S1) is the quotient simplicial group of F (S1) modulo the normal s*
*ub-
simplicial 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 boundaries BF (S1). N*
*otice
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
Corollary 2.13, 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(ss1(Q); *
*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 nwith 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 let W = [[xi1; xi2]; . .;.xit]; xit] be a co*
*mmutator
such that ip 6= iq for p < q. __
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 = x
k+1 k i;
for xk = sn-1 . .^.sk.s.0.oe 2 S1n. Thus
sj[[xi01; xi02]; . .;.xi0t]; xi0t] = [[xi1; xi2]; . .;.xit]; xi*
*t];
where i0k= ik if ik < j and i0k= ik - 1 if ik > j. By induction,
__
[[xi01; xi02]; . .;.xi0t]; xi0t] 2 B :
ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF CERTAIN SPACES 31
__
Thus W = [[xi1; xi2]; . .;.xit]; xit] 2 B . The induction is finished and the_*
*assertion_
follows. |_*
*_|
The simplicial group K(S1) is homotopy equivalent to a product of the Eilenbe*
*rg-
Maclane spaces with a different multiplication.
Proposition 6.15. K(S1) with the loop multiplication is an abelian simplicial *
*group.
Therefore K(S1) is homotopy equivalent to a product of the Eilenberg-MacLane sp*
*aces
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 Ker(d0)\Kn+1(S1) ~= is the normal subgro*
*up__
generated by x0 which is abelian by Lemma 6.4. The assertion follows. *
* |__|
Remark 6.16. Let BK(S1) be the classifying spaces of K(S1). Then BK(S1) is
NOT a product of Eilenberg-Maclane space by Proposition 6.11, where the Whitehe*
*ad
product on ss*(BK(S1)) is given by Proposition 6.11 and so the Gerstenhaber alg*
*ebra
ss*(BK(S1)) is determined. But the cohomology of BK(S1) is still unknown even in
rational case.
In the end of this section, we give some applications of K(S1) to minimal sim*
*plicial
groups. Let G be a simplicial group. We write Gab for the abelianlization of G.
Proposition 6.17. Let G be a minimal simplicial group. If Gab is a minimal sim*
*pli-
cial group K(ss; 1) for a cyclic group ss, Then G is homotopy equivalent to a p*
*roduct
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 representative 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 6.18. The simplicial homomorphism g :F (S1) ! G is simplicial surjectio*
*n.
Proof.It suffices to show that the subsimplicial group, denote by H, of G gener*
*ated
by G1 is G itself. This is given by induction on the dimensions starting with H*
*1 =
G1. Suppose that Hn-1 = Gn-1 with n > 1. By [23, Proposition 1, pp.6], Gn
is generated by the degenerate images of lower order Moore chain complex terms
and NGn. Thus Gn is generated by NGn and Hn by induction. Notice that G is
a minimal simplicial group. Thus NGn = ZGn, the cycles. By Lemma 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
32 JIE WU
Gabn. Notice that Gab = K(ss; 1). Thus N(Gab)n = 1 for n > 1 and so the composi*
*te
ZGn = NGn ! Gn ! Gabnis trivial. Thus the homomorphism OE: ssnG ! Gn=Hn is *
* __
trivial and therefore Gn=Hn is trivial. The assertion follows. *
* |__|
Continuation of the proof of Proposition 6.17.Notice that G is minimal. The si*
*m-
plicial epimorphism g :F (S1) ! G factors through K(S1) by Theorem 6.14. By
Proposition 6.15, K(S1) is an abelian simplicial group. Thus G is an abelian
simplicial group and so G is homotopy equivalent to a product of Eilenberg-Macl*
*ane_
spaces, which is the assertion. *
*|__|
The following counter-example for minimal simplicial groups is due to J. W. M*
*ilnor
(unpublished).
Proposition 6.19. (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*: ss3(F (S1)) = Z=2 ! ss3(n-1G) = Z=2 is an
isomorphism. Notice that n-1G is also a minimal simplicial group. The simplicial
homomorphism f :F (S1) ! n-1G factors through K(S1). Notice that 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 [28]. It is known that there are many counter-examples of two-stage Postnik*
*ov
systems for minimal simplicial groups [28].
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Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104,
USA, jiewu@math.upenn.edu
~~