The Inverses of an H-Space
Martin Arkowitz, Hideaki Oshima and Jeffrey Strom
Abstract A multiplication on an H-space X has a left inverse ~ and a right inv*
*erse æ. They are
mutual inverses and ~ = æ if and only if ~2 = id. In this paper we investigate *
*the order |~| of ~. We
give an example of a multiplication with |~| = 6, and prove that for any finite*
* H-complex X there
are finitely many left inverses of finite order. Conditions are given for there*
* to be infinitely many
multiplications on X with the same left inverse. We then give conditions for a *
*left inverse to have
infinite order. We apply these results to specific Lie groups.
2000 MSC: 55P45, 55P62
Keywords: H-Spaces, homotopy inverse
1 Introduction
An H-space is a pair (X, ~) consisting of a based topological space X and a hom*
*otopy
class ~ 2 [X x X, X], called the multiplication, whose restriction to each fact*
*or of
X x X is id, the homotopy class of the identity map of X. If, in addition, X i*
*s a
(finite) CW-complex, we call X or (X, ~) a (finite) H-complex. An H-space (X, ~*
*) is
group-like if ~ is homotopy associative, i.e., ~(~ x id) = ~(id x ~) 2 [X x X x*
* X, X]
and if ~ has a homotopy inverse. The latter condition is that there exists ' 2 *
*[X, X]
such that ~('xid) = 0 = ~(idx') , where is the homotopy class of the diagonal
map of X and 0 is the constant homotopy class.
For any based space A, a multiplication ~ of X induces a binary operation on
[A, X] defined by ff + fi = ~(ff x fi) such that ff + 0 = 0 + ff = ff. If ~ is*
* homotopy
associative, then [A, X] is associative. If (X, ~) is group-like, then [A, X] i*
*s a group.
A great deal of work has been done on the homotopy associativity condition of
an H-space [St1, St2, Za]. On the other hand, the inverse condition has been st*
*udied
very little. The reasons for this may be the following: (1) the inverse conditi*
*on rarely
appears as a hypothesis in theorems about H-spaces (2) the result of James whic*
*h we
discuss next.
Recall that an (algebraic) loop L is a set with an additive binary operation*
* such
that for every a, b 2 L the equations a + x = b and y + a = b have unique solu-
tions x, y 2 L. Then James has proved [Ja3, Thm. 1.1] that [A, X] is a loop if *
*A is
a CW-complex and (X, ~) is an H-space. Thus every ff 2 [A, X] has a unique left
inverse ffL and a unique right inverse ffR defined by ffL + ff = 0 and ff + ffR*
* = 0.
Hence if (X, ~) is an H-complex, there are unique elements ~, æ 2 [X, X], calle*
*d the
left and right inverse of ~, which are the left and right inverse of id, respec*
*tively, i.e.,
~(~ x id) = 0 = ~(id x æ) . It follows as in group theory that if ~ is homotopy
1
associative, then ~ = æ, and so (X, ~) is group-like. Thus the inverse property*
* auto-
matically holds for homotopy associative H-complexes, and this may be a reason *
*why
this condition has not received much attention. However, many familiar homotopy*
* as-
sociative H-complexes such as compact, connected Lie groups, admit infinitely m*
*any
multiplications which are not homotopy associative [Cu , Thm. II] and thus may *
*have
left and right inverses which are not equal [AL , Cor. 4.4]. It is therefore re*
*asonable
to study the inverse condition for arbitrary multiplications. We do this in thi*
*s paper.
We next give a brief outline of the paper. We let (X, ~) be an H-complex wi*
*th
left inverse ~ and right inverse æ. In x2 it is shown that ~ and æ are homotopy
equivalences and mutual inverses and that ~ = æ if and only if ~ O ~ = ~2 = id.*
* We
construct a class of multiplications on any H-space all of which have the same *
*left
inverse and the same right inverse. In x3 we consider the order |~| of a left i*
*nverse
~ of a multiplication on X, i.e., the smallest positive integer n such that ~n *
*= id;
or 1, if there is no such integer. Since |~| = 2 corresponds to ~ = æ, we regar*
*d the
order of ~ as a measure of how much ~ differs from æ. Since ~ = æ for homotopy
associative multiplications, we could also regard a large order |~| as indicati*
*ng that the
multiplication is highly nonhomotopy associative. In x4 we use methods of ratio*
*nal
homotopy theory to study inverses on an H-space X by investigating multiplicati*
*ons
and inverses on the Sullivan minimal model M of X. We prove that for any finite
H-complex, the set of all left inverses of finite order is a finite set. We ob*
*tain an
easily verifiable condition on a homotopy associative finite H-complex X (in te*
*rms
of the rational cohomology of X) for there to be infinitely many multiplication*
*s with
the same left inverse. This leads to a determination of which 1-connected simpl*
*e Lie
groups have this property. Finally, we give necessary and sufficient conditions*
* for X
to admit a multiplication with |~| = 1, and determine which 1-connected simple *
*Lie
groups have this property.
We conclude this section by giving our notation and terminology. The usual
conventions of homotopy theory will hold. All spaces will be based and will hav*
*e the
homotopy type of CW-complexes. They will be assumed to be nilpotent, and will
often be 1-connected. All maps and homotopies will preserve base points. We will
not distinguish notationally between a map and its homotopy class, but will ref*
*er to
an actual map as a function. For spaces A and X, we let [A, X] denote the set *
*of
homotopy classes from A to X. A map (or homotopy class) f : A ! A0determines
a function f* : [A0, X] ! [A, X] in the obvious way. Furthermore, f induces a
homomorphism of homotopy groups, denoted f# : ßs(A) ! ßs(A0).
2
2 Basic Properties
Let X be an H-complex with multiplication ~, left inverse ~ and right inverse æ*
* and let
A be a based space. If ff 2 [A, X], then it easily follows that ffL = ~ff and f*
*fR = æff.
From ~ + id = 0 = id + æ, we obtain id = ~R = æ~ and id = æL = ~æ.
Lemma 2.1 ~æ = id and æ~ = id. Thus ~ and æ are homotopy equivalences and
mutual inverses. 2
For an element ff 2 [X, X] and positive integer n, let ffn = ffff . .f.f (n *
*times).
The following lemma is then obvious.
Lemma 2.2 ~2 = id () ~ = æ () æ2 = id. 2
Lemma 2.3 For every i > 0 and for every x 2 ßi(X), we have ~# (x) = -x = æ# *
*(x).
Proof. This follows since ~# + id# = (~ + id)# = 0# = 0 : ßi(X) ! ßi(X), and
similarly for æ. 2
Now we construct a class of multiplications related to a given one by commut*
*ators.
Let (X, ~0) be an H-space with left and right inverses ~0 and æ0, respectively.*
* We
define a commutator OE0 2 [X x X, X] as follows. Let p1, p2 2 [X x X, X] be the*
* two
projections and set
OE0 = (p1 + p2) + (æp1 + æp2).
Note that if ~0 is homotopy-associative, OE0 = p1+ p2- p1- p2 = [p1, p2], the c*
*ommu-
tator in the group [X x X, X].
For a fixed multiplication ~0 on X, a positive integer n and an element ff 2*
* [A, X],
define nff inductively by 1ff = ff, 2ff = ff + ff, . .,.nff = (n - 1)ff + ff.
Definition 2.4 Given ~0, ~0 and æ0 as above, define a multiplication ~s on X by
~s = ~0 + (sOE0),
where s 0 is an integer.
Remark 2.5 (1) If ~0 is the standard multiplication on a compact, connected L*
*ie
group X, then James has proved that OE0 has finite order [Ja2, Lem. 4.1]. If k *
*is the
order of OE0, there are k multiplications of the form ~s. Furthermore if X is *
*not a
product of circles, then k 2 [Hu , Thm. 1.1].
(2) Another definition of a commutator element is _0 = (~0p1 + ~0p2) + (p1 + p2*
*) 2
[X x X, X]. When ~0 is homotopy-associative, _0 = -p1 - p2 + p1 + p2. One could
use _0 in place of OE0 to define a class of multiplications analogous to ~s.
3
Proposition 2.6 If ~s and æs are the left and right inverses of ~s, then ~s = *
*~0 and
æs = æ0.
Proof. Let +s denote the binary operation in [X, X] induced by ~s. Consider
~0 +sid = (~0 + sOE0)(~0 x id)
= ~0(~0 x id) + sOE0(~0 x id) .
But ~0(~0 x id) = ~0 + id = 0 and
OE0(~0 x id) = ((p1 + p2) + (æ0p1 + æ0p2))(~0 x id)
= (~0 + id) + (æ0~0 + æ0)
= 0 + (id + æ0)
= 0.
Thus ~0 +sid = 0 + s0 = 0, and so ~0 = ~s. A similar argument shows
id +sæ0 = s(æ0 + æ20).
But æ0 + æ20= (id + æ0)æ0 = 0. Therefore æ0 = æs. 2
Remark 2.7 We have noted that if ~ is homotopy-associative, then ~ = æ. We now
observe that the converse is false. For if ~0 is a homotopy-associative multipl*
*ication,
then ~0 = æ0. Thus for any s < the order of OE0, we have ~s = æs. But ~s need
not be homotopy-associative. One concrete example of this occurs with X = S3
and s = 1, 4, 7 or 10 [AC3 , Rem. 1]. Another example occurs with the exception*
*al Lie
group (G2, ~s). Since ~s = ~0 = æ0 = æs, we show that ~s is not homotopy-associ*
*ative
for s = 1, 4, 7, 10, . ... If i : S3 ,! G2 is the inclusion, then i : (S3, ~s) *
*! (G2, ~s)
is an H-map. Hence i*<'3, '3>s = *s where <-, ->s is the Samelson product *
*with
respect to ~s and '3 is the identity map of S3. By [Sc, Thm. 0.1], if ~s is hom*
*otopy-
associative, then **s is a generator of ß6(G2) = Z=3Z [Mi]. Hence **0 *
*is a
generator of ß6(G2). Since <'3, '3>s = (2s + 1)<'3, '3>0 by [AC3 , Lem. 4], we *
*have that
**s = i*<'3, '3>s = (2s + 1)i*<'3, '3>0 = (2s + 1)**0. Therefore if s *
* 1 (mod 3),
then **s = 0 so that ~s is not homotopy-associative.
3 The Order of Inverses
Let (X, ~) be an H-space with left inverse ~ and right inverse æ. The (multipli*
*cative
or composition) order of ~ is the smallest positive integer n such that ~n = id*
*, and
4
we write |~| = n. If there is no such positive integer, let |~| = 1. Then |~| =*
* |æ| by
Lemma 2.1. If 2ßi(X) 6= 0, for some i, then ~ 6= id since ~# = -1. The condition
2ßi(X) 6= 0, for some i, holds for any non-contractible, finite H-complex [Cl, *
*Thm. 1].
Thus if X is a non-contractible finite H-complex, then |~| = |æ| = 2 if ~ is ho*
*motopy-
associative. More generally, if [X, X] is a group, then |~| = |æ| = 2. In par*
*ticular,
this is the case for every multiplication on S1, S3, S7, Sm x Sn (m, n 2 {1, 3,*
* 7}),
SU(3) and Sp(2).
Now assume that (X, ~) is a non-contractible, finite H-complex, but ~ is not
necessarily homotopy-associative. What is |~|? First note that if |~| < 1, then*
* |~|
is even. For if |~| = n with n odd, then for all x 2 ßi(X),
x = ~n#(x) = (-1)nx = -x.
But since X is a non-contractible, finite H-space, 2ßi(X) 6= 0, for some i. Th*
*is
contradicts x = -x.
We next give an example of a multiplication whose left inverse has order 6. *
*Let
S3 have the standard multiplication ~0 induced from the quaternion structure of*
* R4
and let OE0 2 [S3 x S3, S3] be the commutator. Let pi: S3 x S3 ! S3, i = 1, 2, *
*be the
two projections and let ßj : S3 x S3 x S3 ! S3, j = 1, 2, 3, be the three proje*
*ctions.
Lemma 3.1 (1) [-ß2, ß1] = [ß2, -ß1] = [ß1, ß2] and
(2) [[ß1, ß2], ß3] 2 [S3 x S3 x S3, S3] has order 3.
Proof. (1) Let p : S3 x S3 x S3 ! S3 x S3 project onto the first two factors.*
* Then
for i = 1, 2, we have ßi= pip so
[-ß2, ß1] = [-p2, p1]p.
But [S3 x S3, S3] is a group of nilpotency 2 [Wh , p. 464], and for such gro*
*ups
[na, b] = n[a, b] for any integer n [Ro , Lem. 5.42]. Therefore
[-ß2, ß1] = -[p2, p1]p = [p1, p2]p = [ß1, ß2].
(2) We have [[ß1, ß2], ß3] = flq = q*(fl), where q : S3xS3xS3 ! S9 is the quoti*
*ent map
onto the smash product and fl 2 ß9(S3) is the triple Samelson product <<'3, '3>*
*, '3>
of the identity class '3 2 ß3(S3). But fl has order 3 [Ja1, x3] and q* : [S9, *
*S3] !
[S3 x S3 x S3, S3] is a monomorphism. Therefore [[ß1, ß2], ß3] has order 3. *
* 2
Let S3 have the standard multiplication ~0 and let OE0 : S3 x S3 ! S3 be the
commutator map. If X = S3x S3x S3 and ` : A ! X is the map such that ß1` = `1,
ß2` = `2 and ß3` = `3, then we write ` = (`1, `2, `3).
5
Example 3.2 Define a multiplication ~ on X by
~ = (~0(ß1 x ß1), ~0(ß2 x ß2), ~0(ß3 x ß3) + OE0(ß1 x ß2)),
(Cf. [AL , Prop. 6.2]). This homotopy class can be represented by the functio*
*n m :
X x X ! X given by
m((a, b, c), (a0, b0, c0)) = (a + a0, b + b0, c + c0+ [a, b0]).
Then
~ = (-ß1, -ß2, [ß2, -ß1] - ß3)
and can be represented by the function l : X ! X defined by
l(a, b, c) = (-a, -b, [b, -a] - c).
Then a simple calculation gives
~2 = (ß1, ß2, [-ß2, ß1] + ß3 + [-ß1, ß2])
= (ß1, ß2, [ß1, ß2] + ß3 - [ß1, ß2])
by Lemma 3.1. But [ß1, ß2] + ß3 - [ß1, ß2] = [[ß1, ß2], ß3] + ß3. Therefore
~2 = (ß1, ß2, [[ß1, ß2], ß3] + ß3).
Since the group [X, S3] has nilpotency class 3, it follows that
~4 = (ß1, ß2, 2[[ß1, ß2], ß3] + ß3)
and
~6 = (ß1, ß2, 3[[ß1, ß2], ß3] + ß3).
Thus ~2 6= id, ~4 6= id and ~6 = id by Lemma 3.1. Therefore |~| = 6.
4 Rational Methods
In this section we use methods of rational homotopy theory to investigate the i*
*n-
verses of an H-space. This section is a continuation and further development of
some of the ideas in [AL ]. We begin by summarizing several basic facts about *
*ra-
tional homotopy theory which we shall need. Some references for this material *
*are
[GM ], [Ha ], [HMR ] and [Su ]. If X is a nilpotent, finite complex, then XQ*
* denotes
6
the rationalization or Q-localization of X and ffQ 2 [XQ, YQ] denotes the ratio*
*nal-
ization of ff 2 [X, Y ]. Furthermore, M = MX denotes the minimal algebra of
X (also called the Sullivan minimal model of X) which is a free-commutative, di*
*f-
ferential graded algebra over the rationals Q with decomposable differential wh*
*ose
cohomology is H*(X; Q). If X is a finite H-complex, the differential of M is z*
*ero
and so M = H*(X; Q) = H*(XQ; Q) = (x1, . .,.xk), the exterior algebra on gen-
erators x1, . .,.xk of odd degree. The homotopy category of rational spaces of *
*finite
type is equivalent to the homotopy category of minimal algebras of finite type.*
* In
particular, [X, Y ] corresponds bijectively to [MY , MX ]. If X and Y are H-s*
*paces,
[X, Y ] Hom (MY, MX).
If M = (x1, . .,.xk) is the minimal algebra of a finite H-complex X, where *
*|xi| =
ni is odd, then a multiplication on M is a homomorphism OE : M ! M M whose
composition with the two projections M M ! M is the identity homomorphism.
(We have called OE a multiplication instead of a comultiplication because it co*
*rresponds
to a multiplication on X.) In M M we denote x 1 by x0and 1 x by x00. Then
a multiplication OE on M can be written
OE(xi) = x0i+ x00i+ P (xi),
where P (xi) is a polynomial in x01, . .,.x0k, x001, . .,.x00keach monomial of *
*which con-
tains as a factor at least one of x01, . .,.x0kand at least one of x001, . .,.x*
*00k. Then P
is called the perturbation of OE [AL , Def. 2.1]. Clearly the perturbation det*
*ermines
the multiplication. A left inverse for a multiplication OE on M is a homomorph*
*ism
fl : M ! M such that r(fl id)OE = 0 : M ! M, where r : M M ! M is the
product homomorphism. The right inverse of OE is similarly defined.
Definition 4.1 Let X be a finite H-complex and Ø 2 [X, X]. Then Ø is called a
quasi-inverse of X if Ø induces multiplication by -1 on all homotopy groups of *
*X.
Let M = (x1, . .,.xk) with |xi| = ni odd. A homomorphism fl : M ! M is
called a quasi-inverse of M if fl induces multiplication by -1 on the vector sp*
*ace of
indecomposables. Thus a left or right inverse for any multiplication of X is a *
*quasi-
inverse of X and a left or right inverse for any multiplication of M is a quasi*
*-inverse
of M. The function defined by fl0(xi) = -xi, for all i is called the trivial qu*
*asi-inverse
of M.
Remark 4.2 If fl is a quasi-inverse of M, note that
fl(xi) = -xi+ Q(xi),
where Q(xi) is a rational polynomial in x1, . .,.xk of degree 3, i.e., Q(xi) *
*is a
decomposable element of M.
7
Proposition 4.3 If fl is a non-trivial quasi-inverse of M, then |fl| = 1.
Proof. Let j be the smallest positive integer such that Q(xj) 6= 0. Then fl(x*
*i) = -xi
for all i < j. Thus
fl2(xj) = -fl(xj) + flQ(xj) = xj- 2Q(xj).
Then by induction,
fln(xj) = (-1)n(xj- nQ(xj)).
Therefore |fl| = 1. 2
Definition 4.4 Let X be a finite H-complex and M its minimal algebra. Then defi*
*ne
a function e1 : [X, X] ! Hom (M, M) to be the composition
[X, X] -l![XQ, XQ] -! Hom (M, M)
and a function e2 : [X x X, X] ! Hom (M, M M) to be the composition
[X x X, X] -l![XQ x XQ, XQ] -! Hom (M, M M),
where l is the Q-localization functor. Thus
e1(Ø) = Ø*Q: M = H*(XQ; Q) ! M = H*(XQ; Q)
and e2(~) = ~*Q: M = H*(XQ; Q) ! M M = H*(XQ; Q) H*(XQ; Q).
Proposition 4.5 Let X be a finite H-complex. Then the set of quasi-inverses of*
* X
of finite order is a finite set. In particular, the set of left inverses of mul*
*tiplications
of X of finite order is a finite set.
Proof. Let E-1(X) be the set of quasi-inverses of X and let E-1,f(X) E-1(X)
be those quasi-inverses of finite order. Let Q(M) be the set of quasi-inverses *
*of M.
Clearly the function e1 of Definition 4.4 induces a function ffl1 : E-1(X) ! Q(*
*M)
defined by ffl1(Ø) = Ø*Q. Then ffl1 carries elements of finite order into eleme*
*nts of finite
order. Thus by Proposition 4.3, ffl1(E-1,f(X)) = fl0, the trivial quasi-invers*
*e. But
e1 : [X, X] ! Hom (M, M) is a finite-to-one function by [HMR , Cor. 5.4]. Th*
*us
E-1,f(X) is a finite set. 2
We have seen that several different multiplications may have the same left i*
*nverse,
e.g., the multiplications ~s of Proposition 2.6. But there are finitely many of*
* these
multiplications. We next consider if there exist infinitely many multiplication*
*s with
the same left inverse. We first discuss some preliminaries.
8
Let X be a finite H-complex and M(X) [X x X, X] the set of homotopy classes
of multiplications of X. Let M = (x1, . .,.xk), with |xi| = ni odd, be the min*
*imal
algebra of X and M(M) Hom (M, M M) the set of multiplications of M. Then
the function e2 : [X xX, X] ! Hom (M, M M) of Definition 4.4 induces a function
ffl2 : M(X) ! M(M). If N is a positive integer and OE 2 M(M) has perturbation
P , then we denote by OE(N)2 M(M) the multiplication having perturbation NP . It
has been proved in [AL , Lem. 4.2] that if X is homotopy associative and OE 2 M*
*(M),
then there is a positive integer N and a ~ 2 M(X) such that ffl2(~) = ~*Q= OE(N*
*).
Proposition 4.6 Let X be a finite, homotopy associative H-complex and suppose *
*that
H*(X; Q) = (x1, . .,.xk) with |xi| = ni odd. If some ni= "1nj1+ "2nj2+ . .+."t*
*njt,
where j1 < j2 < . .<.jt, each "s = 1 or 2 and some "s = 2, then there are infin*
*itely
many multiplications on X which have the same left inverse.
Proof. Define a multiplication _ on M with perturbation P as follows: If j 6*
*= i,
set P (xj) = 0; otherwise set
P (xi) = x0j1. .x.0jt(x00j1)a1. .(.x00jt)at,
where as = "s - 1. Then _ is a multiplication on M with left inverse fl0, in f*
*act,
for any positive integer N, _(N) is a multiplication with left inverse fl0. The*
*n there
exists a positive integer N1 and a multiplication ~1 on X such that ffl2(~1) = *
*_(N1).
Now repeat this process with _(N1)replacing _. We obtain a positive integer N2
and a multiplication ~2 on X such that ffl2(~2) = (_(N1))(N2)= _(N1N2). We cont*
*inue
in this way and obtain infinitely many multiplications ~1, . .,.~s, . .o.n X. *
*Let ~s
be the left inverse of ~s. Hence ffl1(~s) = (~s)*Qis a left inverse for _(N1..*
*.Ns). But
Hom(M, M), with binary operation induced by _(N1...Ns), is a loop [AL , Lem. 3.*
*1],
and so ffl1(~s) = (~s)*Q= fl0. But ffl-11(fl0) is finite (see the proof of Prop*
*osition 4.5).
Thus there exists infinitely many positive integers s1, . .,.sn, . .s.uch that *
*~s1= . .=.
~sn = . ...Therefore all the multiplications ~sn have the same left inverse. *
* 2
We note that the condition in Proposition 4.6 is easily checked for any H-sp*
*ace
whose rational cohomology is known. In particular, this is so for the simple L*
*ie
groups. Thus we have
Example 4.7 The following simple Lie groups have infinitely many multiplicati*
*ons
with the same left inverse: SU(n), n 6; Sp(n), n 8; Spin(2n), n = 5, 7 and *
*n 9;
Sp(2n + 1), n 8; E6 and E8.
Finally we give necessary and sufficient conditions for an H-space to admit a
multiplication whose left inverse has infinite order. We let E# (X) [X, X] de*
*note the
group of homotopy classes which induce the identity homomorphism on all homotopy
groups.
9
Proposition 4.8 If X is a finite, homotopy-associative H-space with H*(X; Q) =
(x1, . .,.xk), |xi| = ni, then the following are equivalent:
(1) X admits a multiplication whose left inverse has infinite order.
(2) E# (X) is an infinite group.
(3) Some ni= ni1+ . .+.nir, 1 i1 < i2 < . .<.ir k and r 3.
Proof. The hypothesis of homotopy-associativity is only used for (3) ) (1).
(1) ) (2): Let ~ be a left inverse of infinite order. Then ~2 2 E# (X) and has *
*infinite
order.
(2) ) (3): This is proved in [AC2 , pp. 31-33].
(3) ) (1): Consider the minimal algebra of X, M = (x1, . .,.xk). Let ni = ni1+
. .+.nirand let N be any positive integer. Let OE(N): M ! M M be a multiplicati*
*on
given by (
x0j+ x00j if j 6= i
OE(N)(xj) = x0 00 0 00 00
i+ xi - Nxi1xi2. .x.irif j = i.
Then a left inverse fl(N) to OE(N) is given by
æ
-xj if j 6= i
fl(N)(xj) =
-xi- Nxi1xi2. .x.ir if j = i .
Since
(fl(N))n(xi) = (-1)n(xi- nNxi1xi2. .x.ir),
fl(N) has infinite order.
Now let OE = OE(1)2 M(M) and consider the function ffl2 : M(X) ! M(M). There
is a positive integer N such that OE(N)= ffl2(~) = ~*Qfor some multiplication ~*
* on X.
Thus if ~ is the left inverse of ~, then ~*Qis the left inverse of OE(N). But H*
*om(M, M)
is a loop, and so ~*Q= fl(N). Therefore ~ has infinite order. *
* 2
For completeness, we apply Proposition 4.8 to the simple Lie groups.
Example 4.9 Each of the following simple Lie groups admits a multiplication w*
*hose
left inverse has infinite order: SU(n), n 8; Sp(n), n 14; Spin(2n), n = 7, *
*9, 11,
13 and n 15; Sp(2n + 1), n 14; E6.
10
References
[AC1] Arkowitz, M. and Curjel, C.: On the number of multiplications of an H-sp*
*ace. Topology
2, 205-208 (1963).
[AC2] Arkowitz, M. and Curjel, C.: Groups of Homotopy Classes, Lecture Notes i*
*n Math. 4,
Berlin: Springer-Verlag 1964.
[AC3] Arkowitz, M. and Curjel, C.: Some properties of the exotic multiplicatio*
*ns on the three-
sphere. Quart. J. Math. 78, 171-176 (1969).
[AL] Arkowitz, M. and Lupton, G.: Loop-theoretic properties of H-spaces. Math*
*. Proc. Camb.
Phil. Soc. 110, 121-136 (1991).
[Cl] Clark, A.: On ß3 of finite dimensional H-spaces. Ann. Math. 78, 193-196 *
*(1963).
[Cu] Curjel, C.: On the H-space structures of finite complexes. Comm. Math. H*
*elv. 43, 1-17
(1968).
[GM] Griffiths, P. and Morgan, J.: Rational Homotopy Theory and Differential *
*Forms. Progress
in Math. 16, Boston: Birkhäuser 1981.
[Ha] Halperin, S.: Lectures on Minimal Models, Memoire S. M. F. 9-10 1983.
[HMR] Hilton, P., Mislin, G. and Roitberg, J.: Localization of Nilpotent Group*
*s and Spaces. Math.
Studies 15, New York: North Holland 1975.
[Hu] Hubbuck, J.: On homotopy commutative H-spaces. Topology 8, 119-126 (1969*
*).
[Ja1] James, I.: Multiplications on spheres II. Trans. Amer. Math. Soc. 84, 54*
*5-558 (1957).
[Ja2] James, I.: On Lie groups and their homotopy groups. Proc. Camb. Phil. So*
*c. 55, 244-247
(1959).
[Ja3] James, I.: On H-spaces and their homotopy groups. Quart. J. Math. 11, 16*
*1-179 (1960).
[Mi] Mimura, M.: The homotopy groups of Lie groups of low rank. J. Math. Kyot*
*o Univ. 6,
131-176 (1967).
[Ro] Rotman, J.: An Introduction to the Theory of Groups. Graduate Texts in M*
*ath. 148, New
York: Springer-Verlag 1995.
[Sc] Schiffman, S.: A Samelson product and homotopy-associativity. Proc. Amer*
*. Math. Soc.
7, 189-195 (1978).
[St1] Stasheff, J.: Homotopy associativity of H-spaces I, Trans. Amer. Math. S*
*oc. 108, 275-292
(1963).
[St2] Stasheff, J.: Homotopy associativity of H-spaces II. Trans. Amer. Math. *
*Soc. 108, 293-312
(1963).
[Su] Sullivan, D.: Infinitesimal Computations in Topology. Publ. Math. I. H. *
*E. S. 47, 269-331
(1977).
[Wh] Whitehead, G.: Elements of Homotopy Theory. Graduate Texts in Math. 61, *
*New York:
Springer-Verlag 1978.
[Za] Zabrodsky, A.: Homotopy associativity and finite CW complexes. Topology *
*9, 121-128
(1970).
11
Martin Arkowitz
Dartmouth College
Hanover, NH 03755
USA
E-mail: martin.arkowitz@dartmouth.edu
Hideake Oshima
Ibaraki University
Mito, Ibaraki 310-8512
JAPAN
E-mail: ooshima@mito.ipc.ibaraki.ac.jp
Jeffrey Strom
Dartmouth College
Hanover, NH 03755
USA
E-mail: jeffrey.strom@dartmouth.edu
12
*