The group of homotopy equivalences of
products of spheres and of Lie groups
Martin Arkowitz and Jeffrey Strom
Abstract We investigate the group E# (X) of self homotopy equivalences of a
space X which induce the identity homomorphism on all homotopy groups. We ob-
tain results on the structure of E# (X) provided the p-localization X(p)of X*
* has the
homotopy type of a p-local product of odd-dimensional spheres. In particula*
*r, we
show that E# (X)(p)is a semidirect product of certain homotopy groups ßn(X(p*
*)). We
also show that E# (X)(p)has a central series whose successive quotients are *
*ßn(X(p)),
which are direct sums of homotopy groups of p-local spheres. This leads to a*
* deter-
mination of the order of the p-torsion subgroup of E# (X) and an upper bound*
* for
its p-exponent. These results apply to any Lie group G at a regular prime p.*
* We
derive some general properties of E# (G) and give numerous explicit calculat*
*ions.
2000 Math. Subject Classifications 55P10, 55P60, 55S37
Keywords Group of homotopy equivalences, localization, p-torsion, Lie group*
*s,
regular primes
1 Introduction
The homotopy classes in [X, X] which induce the identity on all homotopy groups*
* form
a group under composition. Although this group, denoted E# (X), has been studied
by many authors [AC2 , Di, DZ , Ma , Ts], there is still a variety of questions*
* that have
not been fully resolved. For example: For which primes p does E# (X) have eleme*
*nts
of order p? What is the order of the p-torsion subgroup Tp(E# (X)) of E# (X)? W*
*hat
is the structure of Tp(E# (X)) _ e.g., what is its exponent? In this paper we *
*make
significant progress on these questions under the assumption that X is an Ho-sp*
*ace,
i.e., a space with the rational homotopy type of a product of odd-dimensional s*
*pheres.
The group E# (X) is a natural subgroup of the group E(X) of all self homotopy
equivalences of X. There are essentially two types of results on E(X) and E# (X*
*): (1)
properties of these groups for large classes of spaces, and (2) detailed calcul*
*ations of the
group structure for specific spaces. Examples of (1) are the theorems that, und*
*er the
assumption that X is a 1-connected finite CW complex, E(X) is a finitely genera*
*ted
group and E# (X) is a finitely generated nilpotent group [Su , Wi , AC1 , DZ ].*
* Examples
1
of (2) include computations of E(X) and, to a lesser extent, of E# (X) when X i*
*s a
complex with few cells [Ru, Sec. 12] such as a product of two spheres [Ru, Sec.*
* 12.3]
and when X is a low dimensional Lie group ([Oka ], [Ru, Sec. 12.4]). The presen*
*t paper
charts a middle course: our main result gives nearly complete information on th*
*e p-
torsion of the group E# (X) in the special case that X is an Ho-space and p > 3*
* is
a regular prime for X. We apply this result both to give general properties of*
* the
p-torsion subgroup of E# (X) and to explicitly calculate the order of Tp(E# (X)*
*) for a
large number of spaces.
If X has the rational homotopy type of a product of odd-dimensional spheres,*
* then
the p-localization X(p)of X has the homotopy type of the homotopy-commutative H-
space P = Sn1(p)x . .x.Snk(p)for all but finitely many primes p. The primes p f*
*or which
X(p) P are called regular for X. If p is regular for X, then E# (X)(p) E# (X(*
*p))
E# (P ) [Ma ], so it suffices for our purposes to study E# (P ).
Certain members of E# (P ) are easy to find: if N = n1+ . .+.nk and q : P -!*
* SN(p)
is the quotient map to the smash product, then the map id+ ffq lies in E# (P ) *
*for
any ff 2 [SN(p), P ] ßN (P ). More generally, for I {1, 2, . .,.k} with |*
*I| 2,
Q ni
let PI = i2IS(p), and write iI : PI ,! P for the inclusion, pI : P -! PI for*
* the
projection and qI : PI- ! SNI(p)for the quotient map. Then id+ ffqIpI 2 E# (P *
*) for
any ff 2 [SNI(p), P ] ßNI(P ). In fact, the set of all such elements forms a*
* subgroup
HI E# (P ) which is isomorphic to ßNI(P ). It is natural to ask to what exten*
*t the
subgroups HI determine the group E# (P ), and what algebraic relations hold bet*
*ween
them. Our main result answers these questions.
Theorems 4.9, 4.10 There is an ordering of the subgroups HI ßNI(P ) such that
there is a semidirect product decomposition
i j
E# (P ) HIm |x HIm-1|x (. .|.x (HI2|x HI1)...).
In addition, there is a central series
{id} Lk Lk-1 . . .L2 = E# (P )
L
for E# (P ) with Ls=Ls+1 |I|=sHI.
Modulo extension problems, these theorems reduce the determination of E# (P ), *
*and
hence E# (X)(p), to the calculation of the homotopy groups of spheres. We obta*
*in
results analogous to Theorems 4.9 and 4.10 for the p-torsion subgroup Tp(E# (P *
*)) of
E# (P ). This yields the following corollary.
Corollary 4.12 Let P = Sn1(p)x . .x.Snk(p). Then
2
Q m
(a) |Tp(E# (P ))| = s=1|Tp(ßNIs(P ))|;
L
(b) if es be the p-exponent of |I|=sßNI(P ), then the p-exponent of E# (P ) *
*is at most
P k
s=2es.
This can be applied to Lie groups.
Theorem 5.5 Let G be a 1-connected compact Lie group.
(a) If p > 1_2dim(G), then E# (G) has no p-torsion.
(b) If, in addition, G is simple and p is a regular prime for G, then the p-ex*
*ponent
of E# (G) is strictly less than the rank of G.
We now give a brief overview of the organization of the paper. After some pr*
*elim-
inaries in x2, we consider in x3 the space P = Sn1(p)x . .x.Snk(p). We make a d*
*etailed
study of the additive subgroup Z# (P ) [P, P ] of homotopy classes which indu*
*ce zero
onLall homotopy groups. We show in Proposition 3.4 that Z# (P ) is isomorphic *
*to
|I| 2ßNI(P ). This result, which may be of independent interest, is used to i*
*nvesti-
gate the group E# (P ) in x4. Because there is a bijection between Z# (P ) and *
*E# (P ),
Proposition 3.4 enables us to write elements of E# (P ) in a canonical form. Fr*
*om this
we deduce Theorems 4.9 and 4.10 and Corollary 4.12. In x5 we consider Ho-spaces
X and prove someQgeneral results before turning to explicit calculations. We se*
*e that
|Tp(E# (X))| = |I| 2|Tp(ßNI(P ))| for all regular primes p. Now ßNI(P ) is a*
* direct
sum of groups ßNI(Sni(p)), and thus we are theoretically able to calculate |Tp(*
*E# (X))|
for any 1-connected Ho-space X and regular prime p. This leads to general resul*
*ts,
such as Proposition 5.3 and Theorem 5.5. We show that, when X is a 1-connected
simple Lie group, the homotopy groups in question are all given by a theorem of*
* Toda
[To ]. Thus the calculation of |Tp(E# (X))| is a purely mechanical operation. *
* Using
the computer algebra system MAPLE for these computations, we have constructed
tables in x5 which give |Tp(E# (X))| at all regular primes p for the classical *
*Lie groups
SU(n) for n = 4, 5, . .,.15, Sp(n) for n = 2, 3, . .,.10, Spin(2n) for n = 4, 5*
*, . .,.10
and all the exceptional Lie groups.
Acknowledgement We would like to thank Ken-ichi Maruyama for several helpful
comments.
2 Preliminaries
In this section we fix our notation and give some general results which will be*
* needed
in the sequel. The usual conventions of homotopy theory will hold.
3
All spaces are based and have the homotopy type of 1-connected CW-complexes.
All maps and homotopies preserve base points. We do not distinguish notational*
*ly
between a map and its homotopy class. For spaces X and Y , we let [X, Y ] deno*
*te
the set of homotopy classes from X to Y . We denote by 0 the homotopy class of *
*the
constant map in [X, Y ] and by idthe homotopy class of the identity map in [X, *
*X].
A map (or homotopy class) f : X -! X0 determines a function f* : [X0, Y ] -! [*
*X, Y ]
in the obvious way. Furthermore, f induces a homomorphism of homotopy groups,
denoted f# : ßs(X) -! ßs(X0). We use ` ' to denote same homotopy type of spaces
and ` ' to denote homeomorphism of spaces or isomorphism of groups.
If S is a finite set, then |S| is the number of elements of S. If, for each *
*j 2 J, Hj
is a subgroup of a group G, then denotes the subgroup generated by *
*the
subgroups Hj. If J = {1, 2, . .,.n}, then H1H2. .H.nis the subset of G consisti*
*ng of
all products h1h2. .h.n, with hi2 Hi. If p is a prime and G a group, then T (G)*
* denotes
the set of elements of finite order in G and Tp(G) denotes the set of elements *
*whose
order is a power of p. If G is a nilpotent group, then T (G) is a normal subgro*
*up of G,
which we call the torsion subgroup of G. If, in addition, G is finitely generat*
*ed then
T (G) is a finite subgroup and Tp(G) is the p-Sylow subgroup of T (G), which we*
* call
the p-torsion subgroup or p-component of G [Ro, Thm.e5.27]. The p-exponent of
G is the least nonnegative integer e such that xp = 1 for all x 2 Tp(G). For gr*
*oups A
and B with A acting on B by homomorphisms, A |x B denotes the semidirect product
of A and B. If A and B are subgroups of a group G, then G = A |x B if G = AB,
A \ B = {1} and B / G. A chain of normal subgroups
{1} C1 C2 . . .Cn = G
is a central series if Ci+1=Ci is contained in the center of G=Ci for each i.
The following group-theoretic lemma will be used in Section 4. The proof is
straightforward and hence omitted.
Lemma 2.1 Let G and N be finitely generated nilpotent groups.
(a) If {1} C1 . . .Cn = G is a central series for G and H G is any subgr*
*oup,
then {1} H \ C1 . . .H \ Cn = H is a central series for H.
(b) If G isQfinite and {1} G1 . . .Gn = G is any chain of subgroups of G, *
*then
|G| = [Gi+1: Gi].
(c) T (G |x N) T (G) |x T (N).
(d) If 1 -! N -! G -! G=N -! 1 is a short exact sequence, then the p-exponen*
*t of
G is at most the sum of the p-exponents of N and G=N.
4
We will also use localization techniques for groups and spaces in this paper*
*. If p
is a prime and G a nilpotent group, then G(p)denotes the p-localization of G. I*
*f X
is a space, then X(p)denotes the p-localization of X. We let Go and Xo denote t*
*he
rationalization of G and X, respectively. The main facts about localization of *
*groups
and spaces, which we shall use freely, can be found in [HMR ].
For any space X, we consider certain subsets of [X, X]. As noted in the int*
*ro-
duction, E# (X) is the group of all homotopy equivalences f : X -! X such that
f# = id: ßs(X) -! ßs(X) for all s > 0. We also consider the subset Z# (X) [X,*
* X]
of all homotopy classes f : X -! X such that f# = 0 : ßi(X) -! ßi(X) for all *
*i.
Elsewhere E# (X) has been denoted E#1 (X) and Z# (X) has been denoted Z#1 (X).
We specialize to group-like spaces Y , i.e., homotopy-associative H-spaces w*
*ith
homotopy inverse. The multiplication on Y induces a group structure on the set
[Y, Y ], which we write additively. Then Z# (Y ) is a subgroup of [Y, Y ].
Proposition 2.2 If Y is group-like, then the function æ : Z# (Y ) -! E# (Y ) *
*defined
by æ(f) = id+ f is a bijection.
Proof
The inverse oe : E# (Y ) -! Z# (Y ) is given by oe(g) = -id+ g. *
* ||
Although Z# (Y ) and E# (Y ) are groups, the function æ need not be an isomo*
*rphism
because the sum in Z# (Y ) is induced from the H-structure on Y and the product*
* in
E# (Y ) is given by composition.
f q 0
Lemma 2.3 Let A -! B -! B [fCA -! A be a cofibre sequence. If Y and Y are
H-spaces and h : Y -! Y 0is any homotopy class, then
h (u + flq) = hu + hflq
for any fl 2 [ A, Y ] and u 2 [B [f CA, Y ].
Proof
The cofibration induces an action of [ A, Y ] on [B [f CA, Y ] for any space Y *
*, which
we denote fl . u. Furthermore, if Y is an H-space, then fl . u = u + flq. Fo*
*r any
homotopy class h : Y -! Y 0, h(fl . u) = (hfl) . (hu) [Hi, Chap. 17]. Thus h(u*
* + flq) =
h(fl . u) = (hfl) . (hu) = hu + hflq. *
* ||
We write T (X1, . .,.Xk) for the fat wedge of the spaces X1, . .,.Xk, i.e., *
*the set
of all elements of X1 x . .x.Xk with at least one coordinate at the base point.
Let j : T (X1, . .,.Xk) -! X1 x . .x.Xk be the inclusion. The quotient X1 x . .*
*x.
Xk=T (X1, . .,.Xk) is denoted by X1 ^ . .^.Xk.
Lemma 2.4 Let q : X1 x . .x.Xk -! X1 ^ . .^.Xk be the quotient map. Then
5
(a) q# = 0 : ßi(X1 x . .x.Xk) -! ßi(X1 ^ . .^.Xk) for all i > 0.
(b) q* : [X1 ^ . .^.Xk, Y ] -! [X1 x . .x.Xk, Y ] has trivial kernel for any s*
*pace Y .
Proof
Part (a) is immediate because j# is surjective. To prove part (b), write R = X*
*1 x
. .x.Xk, S = X1^ . .^.Xk and T = T (X1, . .,.Xk), and consider the following ex*
*act
sequence of sets
q* ( j)*
[R, Yo]o______[S, Yo]o______[ T, Y ]oo______[ R, Y ].
Since ( j)* has a section [Hi, Chap. 11], it is surjective. By exactness, the k*
*ernel of
q* is trivial. *
* ||
In this paper we work with products of p-local odd-dimensional spheres. We
conclude this section by establishing some notation and basic results that we w*
*ill use
in dealing with these spaces.
Let P = Sn1(p)x . .x.Snk(p), where each ni 3 is odd and p > 3 is prime. Eac*
*h Sni(p)is
a homotopy-commutative group-like space [Ad ]. We give P the product H-structur*
*e,
so P is also a homotopy-commutative group-like space.
Let I be the collection of all subsets In= {i1, . .,.ir} of {1, 2, . .,.k} w*
*ith |I| = r
2. For I 2 I, write PI for the subproduct S(i1p)x . .x.Snir(p) P and let iI : *
*PI- ! P
and pI : P -! PI be the inclusionnand projection, respectively. Notice that i*
*I and
pI are H-maps. Let TI = T (S(p), . i1.,.Snir(p)) be the fat wedge contained in *
*PI. Then
PI=TI SNI, where NI = ni1+ . .+.nir. Let qI : PI- ! SNI be the quotient map.
It is essential in many of our proofs to order the set I.
Definition 2.5 An order < on the set I is admissible if whenever I, J 2 I and
I < J then I 6 J.
The following lemma is the reason for our interest in the condition I 6 J.
Lemma 2.6 If I 6 J, then qIpIiJ = 0.
Proof p
The image of PJ -iJ!P -!I PI is contained TI, so qIpIiJ = 0. *
* ||
Finally, we give a particular example of an admissible order that will be us*
*ed in
the sequel.
Example 2.7 For each l = 2, . .,.k let Il be the set of all I 2 I with |I| = *
*l. For
each l choose any order |J|if |I| 6= |J|
I < J if and only if
I 3 is prime. In this section we determine the structu*
*re of
the abelian group Z# (P ) in terms of the homotopy groups of spheres.
For each I 2 I, let ~0I: [SNI(p), P ] -! [P, P ] be the homomorphism define*
*d by
the formula ~0I(ff) = ffqIpI. Since [SNI(p), P ] ßNI(P ) and the map qI#*
* = 0 :
ßs(PI) -! ßs(SNI(p)) for all s > 0 by Lemma 2.4, ~0Idetermines a homomorphism
~I : ßNI(P ) -! Z# (P ).
Lemma 3.1 For each I 2 I, the homomorphism ~I : ßNI(P ) -! Z# (P ) is inject*
*ive.
Proof
The map i*Iis a left inverse for p*I, so p*Iis injective. Therefore ~I = q*Ip*I*
*is injective
by Lemma 2.4. ||
It follows that each I 2 I gives rise to a subgroup of Z# (P ) which is isom*
*orphic
to ßNI(P ).
(3.2) For I 2 I, we write GI = Im(~I) Z# (P ).
Now let Ts P = Sn1(p)x . .x.Snk(p)be the sth intermediate wedge, i.e., the
subcomplex consisting of all k-tuples with at least k - s coordinates equal to *
*the
base point. Thus T1 T2 . . .Tk-1 Tk = P with T1 = Sn1(p)_ . ._.Snk(p)and
Tk-1 = T (Sn1(p), . .,.Snk(p)).
Lemma 3.3 Let f 2 [P, P ] and suppose f|Ts =P0. Then there are maps fI 2 GI *
*for
each I 2 I with |I| = s + 1 such that f|Ts+1= ( |I|=s+1fI)|Ts+1.
Proof
Let fI = fiIpI for each I 2 I with |I| = s + 1. Since f|Ts = 0 we have a commut*
*ative
diagram
____iI_________________________________*
*______________________________________________________________________@
_____________________________________________*
*______________________________________________________________________@
pI ______kI____________________&&__________________*
*____________________________js+1
P __________//_PI__________//Ts+1__________//_P
qI|| qs+1|| |f|
fflffl|~I fflffl|fe fflffl|
SNI _________//_Ts+1=Ts________//P
for each such I, where js+1, kI and ~I are inclusions, qs+1 is the projection a*
*nd ~fis
induced by f. Write ffI = ef~I 2 ßNI(P ). Since fI = ffIqIpI, we have fI 2 GI.
7
P
Let g = |I|=s+1fI. We will show fjs+1 = gjs+1. Since fI|Ts = 0 for each I,*
* we
have g|Ts = 0, and so gjs+1 factors through Ts+1=Ts
js+1
Ts+1 ___________//_P
qs+1|| |g|
fflffl|eg fflffl|
Ts+1=Ts__________//P.
W N e
Observe that Ts+1=Ts |I|=s+1S I so it suffices to show eg|SNJ = f|SNJ for ea*
*ch J
with |J| = s + 1. Using Lemma 2.6 we calculate
X
q*J(eg|SNJ)= eg~JqJ = giJ = ffI(qIpIiJ) = ffJqJ = q*J(fe|SNJ).
|I|=s+1
This proves the result because q*Jis injective by Lemma 2.4. *
* ||
Now we are prepared to prove the main result of this section, the determinat*
*ion
of the structure of Z# (P ).
Proposition 3.4 Let P = Sn1(p)x . .x.Snk(p), where each ni 3 is odd and p > *
*3 is
L L
prime. Then Z# (P ) = |I| 2GI |I| 2ßNI(P ).
Proof
We first show that Z# (P ) is equal to the sum G of the subgroups GI. Let f 2 Z*
*# (P ).
It follows that f|T1 = 0. Assume inductively that there is a map gs 2 G such t*
*hat
(f - gs)|Ts = 0 for somePs 1. By Lemma 3.3, there exist fI 2 GI, |I| = s + 1,
such that ((f - gs) - |I|=s+1fI)|Ts+1 = 0. Thus there is a gs+1 2 G such that
(f -gs+1)|Ts+1= 0. By induction, there is a gk 2 G such that f -gk = (f -gk)|Tk*
* = 0,
so f 2 G.
Choose an admissible ordering on the set I. To finish the proof, we show th*
*at
\ GJ = 0 for each J 2 I. SupposePf 2 \ GJ. Since
f 2 , it can be written f =P I 3 *
*is
prime. In this section we use the results of Section 3 to study the structure o*
*f E# (P ).
The main results express E# (P ) in terms of the homotopy groups of spheres. Fi*
*x an
admissible ordering on the set I. We will often write the elements of I in ord*
*er as
I1, I2, . .,.Im .
For each I 2 I define `0I: [SNI(p), P ] -! [P, P ] by the formula `0I(ff) = *
*id+ ffqIpI.
Proposition 4.1 The function `0I: [SNI(p), P ] -! [P, P ] determines a monomor*
*phism
`I : ßNI(P ) -! E# (P ).
Proof
Since (id + ffqIpI)# = id# + ff# (qI)# (pI)# = id# by Lemma 2.4, the image of*
* `0I
is contained in E# (P ). Therefore we may restrict the range to obtain a map `*
*I :
ßNI(P ) -! E# (P ). To show that `I is a homomorphism we calculate
`I(ff) O `I(fi)= (id+ ffqIpI)(id+ fiqIpI)
= id+ fiqIpI + ffqIpI(id+ fiqIpI)
= id+ fiqIpI + ffqI(pI + pIfiqIpI)
= id+ fiqIpI + ffqI(id+ pIfiqI)pI
= id+ fiqIpI + (ffqI + ffqIpIfiqI)pI
by Lemma 2.3. But (qI)# = 0 by Lemma 2.4, so qIpIfi = 0 : SNI(p)-!SNI(p). There*
*fore
`I(ff) O `I(fi) = id+ fiqIpI + ffqIpI = `I(ff + fi). To show that `I is a monom*
*orphism,
suppose `I(ff) = id. Then ffqIpI = 0. Hence q*I(ff) = 0, and so ff = 0. *
* ||
9
Just as for Z# (P ), each I 2 I gives rise to a subgroup of E# (P ) isomorph*
*ic to
ßNI(P ).
(4.2) For each I 2 I, we write HI = Im(`I) E# (P ).
Thus every element xI 2 HI has a unique expression of the form xI = id+ ffIqIpI
with ffI 2 ßNI(P ). It follows that the bijection æ of Proposition 2.2 restric*
*ts to an
isomorphism of the subgroup GI Z# (P ) with the subgroup HI E# (P ).
P
Lemma 4.3 P If f 2 E# (P ) is represented f = id+ I2IaI as in Corollary 3.5,*
* then
fiJ = iJ + I JaIiJ.
Proof P
By Lemma 2.6, aIiJ = 0 for I 6 J , and so fiJ = iJ + I JaIiJ. *
* ||
For each J 2 I the group E# (P ) acts on the set [PJ, P ] by composition on *
*the
left.TLet Stab(iJ) be the stablizer subgroup of iJ under this action. For K 2 I*
*, let
MK = J>K Stab(iJ). These subgroups form a chain
{id} MI1 MI2 . . .MIm = E# (P )
with MI1= HI1.
P
Lemma 4.4 For each K 2 I, MK = {id+ I KaI| aI 2 GI}.
Proof P P
First we show that Stab(iJ) = {id + I6 JaI|PaI 2 GI}. Write f = id+ IaI and
supposePf 2 Stab(iJ). Then iJ = fiJ = iJ + I JaIiJ by Lemma 4.3. We conclude
that I JaIiJ = 0. Since aI 2 GI, we can write aI = ffIqIpI for some ffI 2 ßNI*
*(P ).
Therefore 0 1
X X X
aI = (ffIqIpI)iJpJ = @ aIiJApJ = 0,
I J I J I J
P P
so f = id+ I6 JaI. Conversely, if f = id+ I6 JaI then fiJ = iJ by Lemma 4.3.
By CorollaryP3.5, a map f 2 E# (X) has a uniqueTrepresentation of the form
f = id+ IaI. We have just proved that f 2 MK = J>KStab (iJ) if and only if
aI = 0 for each I J with J > K. Since the ordering of I is admissible, this *
*is
precisely the statement that aI = 0 for I > K. *
* ||
Proposition 4.5 The group MIs+1is equal to HIs+1MIsfor s = 1, . .,.m - 1.
Proof
Since HIs+1and MIs are both contained in MIs+1,Pit suffices to show that MIs+1
HIs+1MIs. Let f 2 MIs+1and write f = id+ t s+1aIt. Set g = id+ aIs+12 HIs+1.
Then fiJ = giJ for each J > Is by Lemma 4.3. Thus g-1f 2 Stab(iJ). Hence
g-1f 2 MIsand so f 2 HIs+1MIs. ||
10
Corollary 4.6 The group MIsis equal to HIsHIs-1. .H.I1for s = 1, . .m..
Hence E# (P ) = HImHIm-1. .H.I1, or equivalently, E# (P ) = HI1HI2. .H.Im. Our *
*next
goal is to show that MIs+1is a semidirect product of HIs+1and MIs. For this we *
*need
the following result.
Lemma 4.7
(a) HIs+1\ MIs= {id}.
(b) Let J, J1, . .J.s2 I with Jt J for each t (repetitions are allowed).*
* Let
Jt(1), . .,.Jt(l)be exactly those terms in the sequence J1, . .J.swhich ar*
*e equal
to J (in order). If f = xJ1. .x.Js2 E# (P ) with xJt2 HJt, then
fiJ = xJt(1). .x.Jt(l)iJ.
In particular, if none of the Jt is equal to J, then fiJ = iJ.
Proof P
If f 2 HIs+1then f = id+ aIs+1. If f 2 MIsthen f = id+ t saIt. By Corollary 3*
*.5,
f = id. This proves (a).
To prove (b) we first show that if I, I0, J 2 I with I, I0 J and xI 2 HI a*
*nd
xI02 HI0, then 8
< iJ if I, I0 < J
xIxI0iJ = : xI0iJ if I < J, I0 = J
xIiJ if I = J, I0 < J.
This is easily proved using Lemma 4.3 in each of the three cases. Part (b) fol*
*lows
immediately from this formula by induction on the word length. *
* ||
Now we establish some deeper group theoretic properties of the subgroups MI.
Theorem 4.8 The commutator subgroup [E# (P ), MIs+1] is contained in MIs. The*
*re-
fore MIsis a normal subgroup of E# (P ) and
{id} MI1 MI2 . . .MIm = E# (P )
is a central series for E# (P ). Furthermore MIs+1=MIs HIs+1for each s.
Proof
Let f 2 E# (P ) and g 2 MIs+1. We show that the commutator [f, g] lies in MIs
by showing [f, g]iI = iI for all I Is+1. By Corollary 4.6 we can write f = (*
*id +
aIm) . .(.id+aI1) and g = (id+bIs+1) . .(.id+bI1). Assume inductively that [f, *
*g] 2 MIt
for some t s + 1. If t > s + 1, then by Lemma 4.7
[f, g]iIt= (id+ aIt)(id- aIt)iIt= iIt.
11
If t = s + 1, then
[f, g]iIt= (id+ aIt)(id+ bIt)(id- aIt)(id- bIt)iIt= iIt.
In either case, [f, g] 2 Stab(iIt). It follows by induction that [f, g] 2 MIs. *
* ||
As a consequence we derive the desired semidirect product decomposition of E*
*# (P ).
Theorem 4.9 For s = 1, . .,.m - 1, MIs+1is the semidirect product of HIs+1and
MIs. Consequently
i j
E# (P ) HIm |x HIm-1|x (. .|.x (HI2|x HI1). .).
where HI ßNI(P ).
Proof
This follows from Proposition 4.5 and part (a) of Lemma 4.7 since MIsis a normal
subgroup of MIs+1by Theorem 4.8. ||
It was shown in [AL, p. 17] that the rationalization of E# (SU(n)), which is*
* iso-
morphic to the rationalization of E# (S3(p)x . .x.S2n-1(p)) (see Proposition 5.*
*1 below),
is nonabelian for n 18. It follows that E# (P ) need not be an abelian group.*
* This
shows that, in contrast to the direct sum decomposition of Z# (P ) in Propositi*
*on 3.4,
the semidirect product decomposition in Theorem 4.9 is not in general a direct *
*sum
decomposition.
In many applications it is desirable to work with the shortest available cen*
*tral se-
ries. Using an admissible order of the kind described in Example 2.7, we can co*
*nstruct
a central series for E# (P ) that is significantly shorter than the one given i*
*n Theorem
4.9. For each s = 2, . .,.k, let Ls be the product, in order, of the HI with |I*
*| s.
Theorem 4.10 The chain
{id} Lk Lk-1 . . .L2 = E# (P )
L
is a central series for E# (P ), with Ls=Ls+1 ~= |I|=sßNI(P ).
Proof
Let J1, . .,.Jl be the index sets with exactly s elements, in order, so that Ls*
* =
Ls+1HJ1. .H.Jl. Let fi : E# (P ) -! E# (P )=Ls+1be the quotient homomorphism. T*
*hen
fi(Ls) = fi(HJ1)fi(HJ2) . .f.i(HJl).
From Theorem 4.8 we know that fi(HJ1) is in the center of E# (P )=Ls+1. If we c*
*hange
our chosen order on I by rearranging just the Jt, the result is another admissi*
*ble
12
order. The proof of Theorem 4.8 works equally well for this order, and we concl*
*ude
that fi(HJt) is in the center of E# (P )=Ls+1 for each t = 1, . .,.l. It follo*
*ws that
Ls=Ls+1 is in the center of E# (P )=Ls+1.
It remains to identify Ls=Ls+1. We know Ls=Ls+1 fi(HJ1) . .f.i(HJl) and fi*
*(HJt)
HJtby part (a) of Lemma 4.7. We will show that (fi(HJ1) . .f.i(HJt))\fi(HJt+1) *
*= {1}
for each t. We have Ls+1 MJtand corresponding homomorphisms
E# (P )M
fiqqqqq MMMMffM
qqq MM
xxqq fl M&&M
E# (P )=Ls+1______________//_E# (P )=MJt.
Now ker(fl) = fi(ker(ff)) = fi(MJt) = fi(HJ1) . .f.i(HJt). Since ff|HJt+1is inj*
*ective, so
is fl|fi(HJt+1), and so we are done. *
* ||
This result can be thought of as an analog for the group E# (P ) of Proposit*
*ion 3.4
or, more precisely, of Whitehead's result [Wh, p. 467].
In the rest of this section we apply our results to the study of the structu*
*re of the
torsion subgroup T (E# (P )) of the finitely generated nilpotent group E# (P ).
Theorem 4.11 Let T be the torsion subgroup of E# (P ).
(a) The chain of subgroups
{id} T (MI1) T (MI2) . . .T (MIm) = T
with T (MIs+1)=T (MIs) T (HIs+1) and the chain of subgroups
{id} T (Lk) T (Lk-1) . . .T (L2) = T
L
with T (Ls)=T (Ls+1) |I|=sT (ßNI(P )) are both central series for T .
(b) The torsion subgroup T has a semidirect product decomposition
i j
T T (HIm) |x T (HIm-1) |x (. .|.x (T (HI2) |x T (HI1)).). .
where T (HI) T (ßNI(P )).
Proof
Part (a) follows immediately from part (a) of Lemma 2.1 and Theorems 4.8 and 4.*
*10.
Part (b) follows from part (c) of Lemma 2.1 and Theorem 4.9. *
*||
13
Thus, up to extensions, we have reduced the problem of calculating T (E# (P *
*))
to that of determining the homotopy groups of spheres. This yields some useful
information about the torsion subgroup of E# (P ).
Corollary 4.12 Let T be the torsion subgroup of E# (P ). Then
Q
(a) |T | = |I||2T (ßNI(P ))|;
L
(b) if es is the p-exponent of |I|=sßNI(P ), then the p-exponent of E# (P ) *
*is at most
P k
s=2es.
Proof
Part (a) follows immediately from part (b) of Lemma 2.1 and part (b) of Theorem
4.11. Part (b) follows from part (d) of Lemma 2.1 and part (a) of Theorem 4.11.*
* ||
These results will be illustrated by concrete examples in the next section. *
*Before
moving on to the applications, we summarize what we have done. We have found
a semidirect product decomposition and two distinct central series for E# (P ).*
* Our
results give the order of the torsion in E# (P ), the rank of E# (P ) and an up*
*per bound
on the p-exponent of T (E# (P )), but not the full structure of E# (P ).
5 Applications
In this section we apply the results of Section 4 to the class of Ho-spaces, wh*
*ich
includes Lie groups, products of odd-dimensional spheres and Stiefel manifolds.*
* We
give the results of many explicit calculations.
As in the previous sections, we let P = Sn1(p)x . .x.Snk(p), where ni 3 are*
* odd and
p > 3 is prime.
Proposition 5.1 Let X be a finite complex with X(p) P = Sn1(p)x . .x.Snk(p). *
*Then
E# (X)(p) E# (P ).
Proof
Maruyama has shown that if X is rationally equivalent to a product of odd-dimen*
*sional
spheres, then the natural map E# (X) -! E# (X(p)) = E# (P ) is in fact p-locali*
*zation
of the group E# (X) [Ma, Cor. 2.10]. The proposition follows. *
* ||
Corollary 5.2 Let X be a finite complex with X(p) P = Sn1(p)x . .x.Snk(p). T*
*hen
Tp(E# (X)) T (E# (P )).
Proof
If G is a finitely generated nilpotent group, then Tp(G) T (G)(p) T (G(p)). *
*Setting
G = E# (X), we obtain the result. *
*||
14
If X is a finite complex and the rationalization Xo of X has the homotopy ty*
*pe of
a product of rational odd-dimensional spheres Sn1ox . .x.Snko, then X is an Ho-*
*space
and is said to have type (n1, . .,.nk). In this situation X(p) Sn1(p)x . .x.S*
*nk(p)for
all but finitely many primes p; the primes for which the equivalence holds are *
*called
regular for X. Theorem 4.11 and Corollary 5.2 allow us to express Tp(E# (X)) in
terms of the p-components of the homotopy groups of odd-dimensional spheres when
p is a regular prime for X. The following result gives these groups in a large *
*range
[To ].
Theorem(Toda) If p is an odd prime, then
8
< Z=p if k = 2i(p - 1) - 1 with 1 i < p
Tp(ß2m+1+k(S2m+1)) = : Z=p if k = 2i(p - 1) - 2 with 2 i < p
0 otherwise if 0 k 2p(p - 1) - 2
for each m 1.
For a given product of odd-dimensional spheres, Toda's theorem might not pro*
*vide
all of the homotopy groups that appear in the decomposition of Theorem 4.10. Al*
*so,
since the first nonzero p-torsion appears in ß2m+1+2p-3(S2m+1), the group E# (P*
* ) will
have no p-torsion for sufficiently large p. The following proposition explicit*
*ly gives
the primes for which Tp(E# (X)) must be trivial and those for which Toda's theo*
*rem
gives all of homotopy groups that appear in the decomposition of Theorem 4.10.
Proposition 5.3 Let X be a finite complexPof type (n1, . .,.nk) and let p be a*
* regular
prime for X. Let c = min(ni) and N = ki=1ni. Then
(a) Tp(E# (X)) = 0 for all p > N-c+3_2.
p ________
1+ 2(N-c)+5
(b) If p > ___________2, then Toda's theorem determines the quotients in the c*
*entral
series of Theorems 4.8, 4.10 and 4.11.
Proof
Observe that the groups that comprise Tp(E# (X)) are of the form Tp(ßM (P )) wh*
*ere
M N. Part (a) follows because if M - c < 2p - 3, then each of these groups is
trivial. For (b) we simply solve N - c < 2p(p - 1) - 2 for p. *
* ||
It is well known that any finite H-space has the rational homotopy type of a
product of odd-dimensional spheres [K, p. 73]. Therefore our entire discussion *
*applies
to such spaces, including compact Lie groups. The following table summarizes t*
*he
relevant information for the 1-connected simple Lie groups [K, p. 73 and Append*
*ix A].
In each row, the column labeled Min Reg gives a lower bound for the set of regu*
*lar
primes for the given group and the column labeled Max Nonzero gives an upper
15
bound for the set of primes p for which Tp(E# (G)) is not automatically trivial*
* by part
(a) of Proposition 5.3.
________________________________________________________
|_Group___|T|ype_________________|Min_Reg_|Max_Nonzero_|_n2-1
| SU(n) |(|3, 5, . .,.2n - 1) n | |____2n(2n+1) |
| Sp(n) |(|3, 7, . .,.4n - 1) 2n ||______2n(2n-1|)
|_Spin(2n)_|(|3,_7,_._.,.4n_-_5,_2n2-(1)n_-|1)___|2_____|
| G2 |(|3, 11) 7 | |7 |
| F4 |(|3, 11, 15, 23) 13 | |23 |
| E6 |(|3, 9, 11, 15, 17, 23)13 | |37 |
| E7 |(|3, 11, 15, 19, 23, 27,135)9 |61 | |
|_E8______|(|3,_15,_23,_27,_35,_39,347,159)1|13|________ |
Table 1: Type and regular primes for connected simple Lie groups
The apparent omission of Spin(2n + 1) is covered by the fact that Spin(2n + 1)(*
*p)
Sp(n)(p)for all odd primes p [Ha ]. One very important consequence of the resu*
*lts
summarized in this table is that for 1-connected simple Lie groups, Toda's theo*
*rem
yields all of the homotopy groups that appear in the decomposition of Theorem 4*
*.10.
Corollary 5.4 If X is a 1-connected simple Lie group and p is a regular prime *
*for
X, then |Tp(E# (X))| is determined by Toda's theorem.
Proof
One simply checks that, for each group G, the lower bound for the regular prime*
*s is
greater than the lower bound given in part (b) of Proposition 5.3. *
* ||
Our discussion has some immediate applications.
Theorem 5.5 Let G be a 1-connected compact Lie group.
(a) If p > 1_2dim(G), then E# (G) has no p-torsion.
(b) If, in addition, G is simple and p is a regular prime for G, then the p-ex*
*ponent
of E# (G) is less than the rank of G.
Proof
Let G have type (n1, . .,.nk) and let p be a regular prime for G. Part (a) fol*
*lows
immediately from statement (a) of Proposition 5.3, using the fact that if G is *
*a 1-
connected Lie group, then c = 3. By part (a) of Theorem 4.11, Tp(E# (G)) has a *
*central
series with length k - 1. By Corollary 5.4 each of the quotients Tp(Ls)=Tp(Ls+*
*1) is
a direct sum of copies of Z=p, and so has p-exponent at most 1. Thus statement *
*(b)
follows from part (b) of Corollary 4.12. *
* ||
Now we turn to some concrete calculations.
16
Example 5.6 We now apply our results to the study of Tp(E# (SU(5))) for regul*
*ar
primes p. Let Q = S3 x S5 x S7 x S9, so SU(5)(p) Q(p)for any regular prime p.
According to Table 1, we only need to consider the primes 5, 7, and 11. First *
*we
examine the prime p = 11. By Toda's theorem
8
< Z=11 if k = 22, 24
ßk(SU(5))(11)= : Z(11) if k = 3, 5, 7, 9
0 otherwise for k 24,
where Z(11)is the integers localized at the prime 11. The only index set I 2 I *
*for which
ßNI(SU(5))(11)6= 0 is {1, 2, 3, 4}. Thus E# (SU(5))(11) E# (Q(11)) = H{1,2,3,*
*4}
Z=11. Next we consider the prime p = 7. This time Toda's theorem yields
8
< Z=7 if k = 14, 16, 18, 20
ßk(SU(5))(7)= : Z(7) if k = 3, 5, 7, 9
0 otherwise for k 24.
The relevant index sets in I are {2, 4} and {3, 4}. It follows from Theorem 4.1*
*0 that
E# (SU(5))(7) E# (Q(7)) = H{2,4} H{3,4}= Z=7 Z=7. We complete the calculati*
*on
with the prime p = 5. According to Toda,
8
< Z=5 if k = 10, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, *
*24
ßk(SU(5))(5)= : Z(5) if k = 3, 5, 7, 9
0 otherwise for k 24.
The relevant index sets are {1, 3}, {2, 3}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, *
*{1, 3, 4},
{2, 3, 4} and {1, 2, 3, 4}. Since there are nine index sets, we see right away*
* that
|T5(E# (SU(5)))| = 59. By Theorem 4.10 we have a central seriesL{id} L4
L3 L2 = (E# (SU(5)))(5)with L4 = H{1,2,3,4} Z=5, L3=L4 |I|=3HI (Z=5)3
L 5
and L2=L3 |I|=2HI (Z=5) . Hence the 5-exponent of E# (SU(5)) is at most
3. Finally we observe that each of these three calculations implies that the g*
*roup
E# (SU(5)) is rationally trivial. Therefore, we have
E# (SU(5)) A2 A3 B5 (Z=7)2 Z=11
where A2 is a finite 2-group, A3 is a finite 3-group, and B5 is a 5-group with *
*order 59
and 5-exponent at most 3.
When the homotopy groups in question are determined by Toda's theorem (as is
the case when X is a 1-connected simple Lie group and p is a regular prime), the
calculation of the order of Tp(E# (X)) and the rank of E# (X) is a purely mecha*
*nical
operation. We have written a MAPLE program which can perform these calculations
for any 1-connected simple Lie group G, though the amount of time required for *
*the
calculation increases with the rank of G.
17
Example 5.7 Tables 2, 3 and 4 give the data from many such calculations, incl*
*uding
the complete calculation of |Tp(E# (G))| for certain Lie groups G and all of th*
*eir regular
primes p.
The order of Tp(E# (G)) is a power of p, say pe. In these tables, the entry*
* in
the row labeled G and the column labeled p is the exponent e. Thus, for exampl*
*e,
the group T31(E# (E8)) has 3135 elements. The entry in the row labeled G and t*
*he
column labeled Rank is the rank of the group E# (G). We use our method to obtain
these numbers, but they have been known for some time [AC1 ]. The notation `zer*
*o'
in an entry means that the group in question is trivial as a consequence of par*
*t (a)
of Proposition 5.3. An entry labeled `0' means that the group was calculated to*
* be
trivial. The notation `_' means that the prime is not regular for the group.
The first table gives the values for p between 5 and 41.
18
_______________________________________________________________________________*
*___
| | | |______________________________Prime___________________________*
*___
|_Group__|_|Rank_|5__|7___|11__|13___|17____|19____|23___|29___|_31___|37___|41*
*___|_
|_SU(4)___|0|____|2__|0___|zero_z|ero_ze|ro__ze|ro__ze|ro_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_SU(5)___|0|____|9__|2___|1___|zero__z|ero__ze|ro__ze|ro_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_SU(6)___|0|____|__|_12__|4___|4____|0_____|zero___z|ero_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_SU(7)___|0|____|__|_43__|15__|12___|6_____|3_____|1____|zero___|zero_ze|ro_ze*
*|ro_ |
|_SU(8)___|1|____|__|____|49__|_37___|23____|17____|8____|2____|_0____|zero__z|*
*ero_ |
|_SU(9)___|2|____|__|____|144_|_110__|67____|52____|33___|19___|_12___|1____|ze*
*ro_ |
|_SU(10)__|4|____|__|____|375_|_291__|184___|145___|93___|69___|_59___|22___|10*
*___|
|_SU(11)__|7|____|__|____|913_|_722__|482___|392___|259__|175__|_165__|110__|69*
*___|
|_SU(12)__|1|1____|__|___|___|__1699_|1180__|988___|695__|425__|_387__|318__|25*
*4__|
|_SU(13)__|1|6____|__|___|___|__3874_|2766__|2369__|1752_|1109_|_964__|733__|65*
*4__|
|_SU(14)__|2|3____|__|___|___|______|_6243__|5428__|4163_|2821_|_2471_|1710_|14*
*55_|
|_SU(15)__|3|1____|__|___|___|______|_13810_|12117_|9525_|6798_|_6076_|4284_|34*
*15_|_
|_Sp(2)____|0|____|1__|zeroz|eroze|ro_ze|ro__ze|ro__ze|ro_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_Sp(3)____|0|____|__|2___|zero_z|ero_ze|ro__ze|ro__ze|ro_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_Sp(4)____|0|____|__|___|2___|_1____|0_____|zero___z|ero_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_Sp(5)____|0|____|__|___|8___|_4____|1_____|0_____|0____|zero___|zero_ze|ro_ze*
*|ro_ |
|_Sp(6)____|0|____|__|___|___|__14___|5_____|3_____|0____|1____|_1____|1____|ze*
*ro_ |
|_Sp(7)____|0|____|__|___|___|______|_18____|12____|7____|2____|_3____|6____|6_*
*___|
|_Sp(8)____|0|____|__|___|___|______|_63____|48____|33___|17___|_12___|16___|21*
*___|
|_Sp(9)____|0|____|__|___|___|______|______|_156___|112__|84___|_71___|41___|46*
*___|
|_Sp(10)___|0|____|__|___|___|______|______|______|_332__|249__|_234__|156__|11*
*7__|_
|_Spin(8)__|0|____|__|7___|0___|0____|zero___z|ero__ze|ro_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_Spin(10)_|0|____|__|___|6___|_4____|3_____|2_____|zero__z|ero__z|ero_ze|ro_ze*
*|ro_ |
|_Spin(12)_|0|____|__|___|16__|_8____|2_____|0_____|0____|2____|_1____|zero__z|*
*ero_ |
|_Spin(14)_|1|____|__|___|___|__55___|33____|25____|17___|10___|_6____|2____|0_*
*___|
|_Spin(16)_|0|____|__|___|___|______|_52____|41____|30___|11___|_13___|26___|26*
*___|
|_Spin(18)_|2|____|__|___|___|______|_221___|184___|130__|78___|_69___|48___|39*
*___|
|_Spin(20)_|0|____|__|___|___|______|______|_419___|319__|252__|_219__|125__|13*
*1__|_
|_G2______|0|____|__|_1___|zero_z|ero_ze|ro__ze|ro__ze|ro_ze|ro__z|ero_ze|ro_ze*
*|ro_ |
|_F4______|0|____|__|____|___|__5____|1_____|1_____|0____|zero___|zero_ze|ro_ze*
*|ro_ |
|_E6______|1|____|__|____|___|__23___|8_____|11____|10___|5____|_0____|zero__z|*
*ero_ |
|_E7______|0|____|__|____|___|______|______|_21____|11___|5____|_4____|2____|3_*
*___|
|_E8______|0|____|__|____|___|______|______|______|_____|_____|__35___|18___|21*
*___|
Table 2: |Tp(E# (G))| for simple connected Lie groups G and 5 p 41
19
The second table gives the values for p between 43 and 97. We have removed ro*
*ws
that consist only of zeros.
________________________________________________________________________________
| Group |_|___________________________Prime____________________________________
|_________|4|3___|47___|53___|59___|61___|67___|71___|73___|79__|83__|_89__|97_*
*_|_
|_SU(10)__|6|___|2____|zero__z|ero_ze|ro__z|ero_ze|ro_ze|ro_ze|roze|ro_z|eroze|*
*ro_|
|_SU(11)__|5|3___|31___|7____|1____|zero__z|ero_ze|ro_ze|ro_ze|roze|ro_z|eroze|*
*ro_|
|_SU(12)__|2|20__|160__|81___|26___|17___|4____|0____|zero__z|eroze|ro_z|eroze|*
*ro_|
|_SU(13)__|6|12__|521__|368__|205__|159__|66___|32___|21___|3___|1___|_zero_z|e*
*ro_|
|_SU(14)__|1|382_|1274_|1092_|817__|712__|423__|278__|220__|92__|43__|_10__|0__*
*_|
|_SU(15)__|3|115_|2761_|2540_|2266_|2125_|1609_|1254_|1088_|657_|428_|_190_|49_*
*_|_
|_Sp(7)____|5|___|3____|zero_ze|ro_ze|ro__z|ero_ze|ro_ze|ro_ze|roze|ro_z|eroze|*
*ro_|
|_Sp(8)____|2|2__2|1___|12___|4____|2____|0____|zero__z|ero_ze|roze|ro_z|eroze|*
*ro_|
|_Sp(9)____|5|2__6|1___|57___|38___|30___|11___|4____|2____|0___|0___|_zero_z|e*
*ro_|
|_Sp(10)___|1|12_1|26__|147__|134__|123__|79___|50___|37___|12__|4___|_1___|1__*
*_|_
|_Spin(14)_|0|___|zero_ze|ro_ze|ro_ze|ro__z|ero_ze|ro_ze|ro_ze|roze|ro_z|eroze|*
*ro_|
|_Spin(16)_|2|3__1|3___|2____|0____|zero__z|ero_ze|ro_ze|ro_ze|roze|ro_z|eroze|*
*ro_|
|_Spin(18)_|3|6__3|0___|19___|12___|11___|5____|3____|2____|zero_z|ero_z|eroze|*
*ro_|
|_Spin(20)_|1|46_1|69__|159__|104__|84___|31___|11___|6____|1___|1___|_1___|zer*
*o_|_
|_E7______|4|___|4____|5____|2____|2____|_zero__z|ero_ze|ro_ze|roze|ro_z|eroze|*
*ro_|
|_E8______|1|4___|19___|13___|10___|5____|5____|17___|5____|4___|14__|_8___|5__*
*_|
Table 3: |Tp(E# (G))| for simple connected Lie groups G and 43 p 97
Only the groups SU(15), Sp(10) and E8 have nontrivial p-torsion for p > 89. T*
*he
values are recorded in the following table.
___________________________________
| Group |_|________Prime__________
|________|1|01_|103_|107_|109_|113_|_
|_SU(15)_|2|1__|13__|3___|1___|zero__|
|_Sp(10)__|1|__|1___|zeroze|roze|ro__|
|_E8_____|1|__|4___|0___|2___|0___|_
Table 4: |Tp(E# (G))| for simple connected Lie groups G and 101 p 113
We have found upper bounds for the p-exponent of E# (X) but we have yet to *
*see
examples of spaces for which this exponent is greater than 1. Our next example*
* fills
this gap.
Example 5.8 According to a theorem of Gray [G ], there are elements of order*
* pt
in the group ß2t+1+2pt-1(p-1)(S2t+1) provided p > 3 and t 1. Since the group
t-1(p-1)-1 pt-1(p-1)+1 2t+1
Tp(E# (Sp x S x S )) contains H{1,2,3}which in turn contai*
*ns
ß2t+1+2pt-1(p-1)(S2t+1), we see that there are elements of arbitrarily large p*
*rime power
order in E# (X), even when X is a product of three spheres. For example, there*
* are
elements of order 73 in the group E# (S293x S295x S7).
20
Even if a prime p is not regular for a space X, it may happen that X(p)has p*
*-local
odd-dimensional spheres as factors. In this situation our results give lower bo*
*unds for
the order of Tp(E# (X)).
Example 5.9 Recall that if X is a Lie group then a prime p is called quasi-re*
*gular
for X if X(p) B(p)x Sn1(p)x . .x.Snk(p)for certain spaces B [MT, p. 325]. We *
*then
call the sequence of odd integers (n1, . .,.nk) the quasi-regular p-type of X. *
*The
following table gives the quasi-regular p-types of some 1-connected simple Lie *
*groups
[MT, Sec. 5]
__________________________________________
|_Group_|_|Prime_|Quasi-regular_p-type_____n |
|_SU(n)__|p|>__2_|(2(n_-_p)_-,1._.,.2(p_+_1) -)1|
|_F4_____|1|1____(|11,_15)________________ |
|_E6_____|1|1____(|9,_11,_15,_17)_________ |
| E7 |1|1 (|11, 19, 27) |
| E7 |1|3 (|15, 19, 23) |
|_E7_____|1|7____(|11,_15,_19,_23,_27)____ |
| E8 |1|7 (|23, 39) |
| E8 |1|9 (|15, 27, 35, 47) |
| E8 |2|3 (|23, 27, 35, 39) |
|_E8_____|2|9____(|15,_23,_27,_35,_39,_47)_ |
Table 5: Some quasi-regular types of Lie groups
Using this information, we obtain the following lower bounds for the order of T*
*p(E# (X)).
The notation `_' means that the prime is not quasi-regular; the notation `reg' *
*means
that the prime is regular.
_____________________________________
| Group |_|________Prime___________
|________|1|1_|13___|17_|19_|23_|29_|_
|_SU(14)_|7|5_|1751_|regr|egr|egr|eg_|
|_SU(15)_|3|1_|778__|regr|egr|egr|eg_|_
|_F4_____|0|_|reg___|regr|egr|egr|eg_|
|_E6_____|1|_|reg___|regr|egr|egr|eg_|
|_E7_____|4|_|2____|2__|reg_|regr|eg_|
|_E8_____|_||_____|_0__|5__|1__|0__|_
Table 6: Lower bounds for |Tp(E# (G))| at quasi-regular primes
Primes not in the table are either not quasiregular or else the lower bound is *
*0.
Remark 5.10 (a) Some authors have studied the related group E*(X) of those *
*self
homotopy equivalences of X that induce the identity on integral homology [Oka ,*
* Wi ,
Z]. If X is a finite H-space whose integral cohomology is torsion-free and prim*
*itively
generated, then it can be shown that E# (X) E*(X), and so all of the calculat*
*ions
in this section give lower bounds for the order of the p-torsion of the group E*
**(X).
21
(b) We have shown in Proposition 3.4 that Z# (P ) is a direct sum of the ßNI(P*
* ).
Thus Tables 2, 3 and 4 actually give the structure of Tp(Z# (G)): if e is the e*
*ntry in
the row labeled G and the column labeled p, then Tp(Z# (G)) (Z=p)e.
(c) Our methods can be used to obtain information on the order and exponent of
the p-torsion subgroup of E# (X) for many other spaces X, including some Stiefel
manifolds and non-simply-connected simple Lie groups.
(d) We emphasize that our method applies to any 1-connected simple Lie group. *
*To
illustrate this, we choose (almost at random) the group SU(97) and the prime p *
*= 137
and mention that
|T137(E# (SU(97)))| = 13755,925,761,774,774,826,517,801,505,265.
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Martin Arkowitz Jeffrey Strom
Dartmouth College Dartmouth College
Hanover NH, 03755 Hanover NH, 03755
martin.arkowitz@dartmouth.edu jeffrey.strom@dartmouth.edu
23