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LIBRARY '
Michigan Stam
University

    
   

This is to certify that the

thesis entitled
7WM 4214 7’7’fl/mow @/ 96W flaw/C
presented by
Ka/i /<. .84.;

has been accepted towards fulfillment
of the requirements for

fiéD 41331331,] ”Mi/MZM

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K. C /U '
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Date KIA/7,

0-7639

 

 

ABSTRACT
TORSION IN H-SPACES OF Low RANK
By

Kai Kie Dai

Our discussion concerns a path connected H-space
X that is also a finite CW complex. By a theorem
of HOpf the reduced cohomology of X ‘with rational
coefficients is an exterior algebra on odd dimensional
generators. The number of generators is called the
£g2§_of X. If Hi(X;Z) contains an element of order
pr(r 2 l), for some i and some prime p, then X

is said to have p-torsion. In particular, if r > 1,

 

then X is said to have highergp-torsion. It has
been shown by W. Browder that if X has rank one, then
X has no p-torsion for any odd p. In'addition, it is
a long standing conjecture that OX, the loop space

of X, is torsion free.

The principal results of this thesis are the
following:
(1) If X has rank two, then X has no

p-torsion for p 2_5.

(2)

(3)

(4)

(5)

(6)

ii

If X is l-connected and has rank 2,
then X has no p-torsion for pig 3.

If X has rank 2, then 0X is torsion
free and X has no higher p—torsion for
any prime p.

If X has rank 2 and has no 2-torsion,
then X has 3-torsion if and only if
H2(X;Z) has an element of order 3.

(a) If X has rank less than or equal
to 5 and has no 2-torsion, then X

has no p-torsion for p > 5.

(b) If X is l-connected, has rank
less than or equal to 4, and has no
2-torsion, then X has no p-torsion

for p > 3.

If X is associative (i.e., there
exists a multiplication that is
associative), l-connected, and has

rank less than or equal to 5, then

X has no p-torsion for p > 5.

TORSION IN H-SPACES OF LOW RANK

BY

Kai Kie Dai

A Thesis

Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1971

ACKNOWLEDGEMENTS

I would like to express my gratitude to my
thesis advisor, Ronald C. O'Neill, for his patient
guidance and enthusiastic encouragement throughout

the work in the thesis and my graduate career.

I also would like to thank the department
of mathematics of Dartmouth College for their
hospitality and for providing me all kinds of

facilities during my visit there in 1970-71,

when this thesis was written.

My thanks and appreciation go to Mary Starr

for the excellent typing of this thesis.

£1 7' / 770

Chapter I

TABLE OF CONTENTS

IntrOduCtion O O O O C O O O O O O O O O 0

Chapter II

Preliminaries and Statements of Results .

§l. On the structure of Hopf algebras . .
§2. On the types of a connected finite
H-complex . . . . . . . . . . . .
§3. On a relation between generators in
H*(X;Zp) and types of X . . . .
§4. Some facts about torsion in H-spaces.
§5. On a relation between the homology
of X and the homology of OX.
§6. On localization of H—complexes. . . .
§7. The cohomology of covering spaces
of H-spaces . . . . . . . . . . .
§8. Statements of results . . . . . . . .
Chapter III
Proofs of Results. . . . . . . . . . . . .

Bibliography

10

14
15

l6
l7

19
20

25

42

CHAPTER I

Introduction

A tOpological space X is an H-space if and
only if there exist a distinguished base point e E X,

the unit element, and a continuous map m : X x X 4 X,

 

called the multiplication, such that m(x,e) = m(e,x) = x
for all x E X. Lie groups and t0pologica1 groups are
H-spaces. There are H-spaces that are not topological

groups; e.g., S7. A finite H-complex is an H-space

 

that is also a finite CW complex (for the definition
of CW complex, see [41; p.401]). A classical

theorm of H0pf says that if X is a path connected
finite H-complex, then its reduced cohomology with
rational coefficients is an exterior algebra on odd

dimensional generators:

H*(X:Q) 3- Mxn ""‘Xn ).

l i
where dim xn = nj = odd. If H*(X;Q) is generated
. j
by xnl,...,xni in dimensions nl,...,ni , respectively,

with nj g_nj+1 , then X is said to have rank i and

type (n1....,ni). If the integral cohomology of X,

H*(X), contains an element of order pr(r'2 l) for
some prime p, then X is said to have prtorsion.
In particular, if r > 1, then X is said to have

higherqp-torsion.

The extension to finite H-complexes of known
topological properties of compact Lie groups provides
the principal motivation for the study of finite
H—complexes. A good example of such a theorem is
the Hopf theorem quoted above. In Hopf's original
paper [28] essentially only Lie groups were treated,
but the paper gave rise to the concept of H-complex
since a continuous multiplication was the only
prOperty required. Prior to the paper of Hilton-
Roitberg [27] it was a standing conjecture that if
X is a finite H-complex, then there exists a Lie
group Gx such that X has the homotOpy type of
OX x S7 x...x S7. However, this conjecture was
demolished by P. Hilton and J. Roitberg by showing
that the 10-manifold M7w , the total space of the
principal 83-bundle over S7 classified by 7w,
where m classifies Sp(2) 4 S7, is an H—space
but not of the homotopy type of any Lie group. Thus

the classification problem of finite H—complexes

will not reduce to that of Lie groups. Precisely,

finite H-complexes are classified up to H-equivalence:

 

An H—map of two H—spaces X and Y with multipli-
cations m and n, respectively, is a map h : X 4‘Y
such that the following diagram commutes up to

homotOpy:

Xxx—Lax

i” i.

YxY-—-n——>Y.

Two H—spaces X and Y are H-equivalent if there exists

 

a homotopy equivalence h : X alY which is an H-map.

One approach to the classification problem
is to investigate how much such a complex must look
homologically like a Lie group, i.e., to investigate
all possible finite H—complexes up to H-equivalence by

their rank.

If X is a path connected finite H-complex
of rank one, then by the result of W. Browder [15], X
has the homotOpy type of SI, 83, S7, RP3, or RP7. In
[38] it is shown that the set of homotopy classes of
multiplications on a finite H-complex X is in one-one

correspondence with [X A X,X], where X A X means the

smash product of X, i.e., the identification space

obtained from X x X by identifying X x [e] U {e} x X
to a single point, and [X A X,X] means the set of

homotopy classes of maps from X A X to X. This set

could be infinite. For 81 there is only one H—space

structure since [S1 A 81,81] 2 [S2

.51] 2W2(Sl) = 0.
There are 12 and 120 homotopy classes on S3 and
S7, respectively. Since each multiplication is H-
equivalent to its transpose or opposite, the number
of non-equivalent H-structures is 6 or 60,

respectively. For the projective spaces the number

of non-equivalent H-structures has not yet been settled.

If X is a connected finite H-complex of

rank two and if X is torsion free, then X has

the homotopy type of S1 x 81, S1 x S3, S1 x 83,

3 3 3 7
S x S , SU(3), S x S , Sp(2), EZw , E3w o 4w , ESw o
7 7

6w . E7w , or S x S . This was proved independently

E
E
by P. Hilton and J. Roitberg [26], M. Curtis and G.
Mislin, E. Thomas, and A. Zabrodsky [unpublished].
The number of homotopy classes of multiplications on
products of spheres can easily be computed and the
number of homotopy classes of multiplications on

SU(3) and Sp(2) is 215 - 39 - 5 ° 7 and

220 - 3 - 55 . 7, respectively [34].

The classification of arcwise connected finite
H-complexes of rank two with torsion is not yet
complete. In fact, it is an open problem that if X
is a l-connected finite H-complex with or without
2-torsion, then X has no p-torsion for p > 5. A
related problem is a long standing conjecture that if
X is a l-connected finite H—complex, then 0X, the

loop space of X, is torsion free.

Torsion plays an important role in homotopy
classification of finite H-complexes. The purpose
of this thesis is to solve some of the problems on

torsion in finite H-complexes.

CHAPTER II

Preliminaries and Statements of Results

The requisite background for the proofs of the
results in this thesis is sketched in the first seven
sections and the principal results of the thesis are

given in the last section.

§l. On the structure of Hopf algebras.

A good reference for this section is [33].

Let R be a commutative ring with unit. A

graded R algebra consists of a graded R module

 

A = {Ag} and a homomorphism of degree 0
u : A ®.A 4.A

called the product of the algebra (u then maps
Ap a,Aq into Ap+q for all p and q). For a,
a' E A we write aa' = u(a m a'). The product is

associative if (aa')a" = a(a'a") for all

 

a,a',a" E A and is commutative if aa'

 

a'a for all a,a' E A.

d I
(-1) eg a deg a

A graded R coalgebra consists of a graded R

module A = {Ag} and a homomorphism of degree 0
d : A 4.A ®.A

called the coproduct of the algebra (so d maps

 

Aq into e. A1 Q.AJ for all q). The c0product

1+j=q

is said to be associative if

 

(d®l)d=(l®d)d:A-¢A®A6¢A

and is said to be commutative if Td = d, where

 

T : A ®.A 4.A ® A is the homomorphism T(a ® a') =
(-1)deg a deg a a' ® a. A connit for the coalgebra
is a homomorphism e : A 4 R such that each of the

composites in the diagram,

R®A

d V i‘
A.
gs

Z\———4> A.8>A
%A®R

is the identity map.

A Hopf algebra over R is a graded R algebra

 

A which is also a coalgebra whose c0product
d z A 4.A ®.A

is a homomorphism of graded R algebras. A HOpf algebra

A is said to be connected if A0 is the free R module

 

generated by a unit element 1 for the algebra and the
homomorphism e : A 4 R defined by a(al) = a for

a E R is a connit for the coalgebra.

Let X be a connected finite H—complex with
multiplication m and K a field. It is well-
known (cf. [36, p.49]) that H*(X;K) is a connected
HOpf algebra with product induced by the multiplication
and the coproduct induced by the diagonal map. Dually,
H*(X;K) is a connected Hopf algebra with product
induced by the diagonal map and coproduct induced by

the multiplication.

If A is a graded coalgebra with coproduct
d : A 4.A ®.A, then an element y of A which is

not in A0 is called primitive if d(y) = y®l + 13y.

 

Thus for any space Y, whether Y is an H-space
or not, an element y in H*(Y;K) is called primitive
if and only if the homomorphism A* induced by the

diagonal map has the following property:

A*(Y) = Y ® 1 + 1 ® Yo

Primitivity of y in H*(Y;K) has nothing to do with
the multiplication. Also if X is a finite H-complex

with multiplication m, then an element y in

H*(X;K) is called primitive if and only if the
homomorphism m* induced by the multiplication m

has the following prOperty:
m*(y)=l®y+y®l.

Notice that primitivity of y in H*(X;K) depends

on the multiplication m.

An element in a graded algebra is called
indecomposable if and only if it cannot be written as

a product of lower nonzero dimensional elements.

Let X be a connected finite H-complex and
K be a field. For simplification let A be either
H*(X;K) or H*(X;K) and A* be its vector space dual.

Then we have:

Lemma 1.1. [33; § 3]. Let P(A) denote the
subspace of primitive elements of A and Q(A) the
subspace of indecomposable elements of A. Then

P(A*) a- (Q(A))*.

Lemma 1.2. [33; § 7]. If K is perfect,
then A is isomorphic as an algebra with a tensor
product A1 ®...®.An, where Ai , 1.3 1.3 n 18 a

Hopf algebra with a single generator Xi'

10

Lemma 1.3. [33; § 7]. If A is associative
and if K of characteristic zero, then A is isomorphic

to an exterior algebra on odd dimensional generators.

Lemma 1.4. [33; § 4]. Suppose that the field
K is of characteristic p # 0. Then there is an exact

sequence:
0 4 P(K(§A)) 4 P(A) 4 Q(A).

where P(A) denotes the subspace of primitive elements
of A, Q(A), the subspace of indecomposable elements
of A, and EA, the image of the homomorphism

. ' _ P .
E . An a'Apn defined by §(x) — x for all x E An'
K(§A) is the subalgebra generated over K by 5A.

In other words, if a primitive element of A is

decomposable, then it is a pth power.

§2. On the types of a connected finite H-complex.

Let X be a connected finite H-complex. There
is a relationship between torsion in X and the possible
types of X. The results listed in this section will
be used to determine the existence of certain p-torsion

in X.

11

Lemma 2.1. [1]. Let X be a connected finite
H—complex of rank one, then X has type (1), (3), or

(7).

Lemma 2.2. [29; Theorem 1.1]. Let X be a

 

connected finite H-complex of rank 2 having no 2—
torsion. Then the type of X is (1,1), (1,3), (1,7),

(3,3), (3,5), (3,7), or (7,7).

We remark here that Lemma 2.2 was first proved
partially by J. Adams [4] and completed by Douglas-
Sigrist [24] and was also proved independently by

J. Hubbuck [29].

Lemma 2.3. [29; Theorem 1.1]. Let X be a
connected finite H-complex of rank less than or equal
to 5 having no 2-torsion. Then the type of X is
a union of sets taken from (1),(3),(7),(3,5),(3,7),
(3,5,7),(3,7,11),(3,5,7,9),(3,7,11,15),(3,5,7,9,11),
or (3,7,11,15,19). I

The method in proving Lemma 2.3 is by studying
the projective plane of the finite H-complex defined
in [43] and using a result by W. Browder and E. Thomas

on the cohomology of the projective plane of X [18].

12

The machinary is K-theory and Adams operations on K—
theory, which, when a space has no torsion, "represent"
the Steenrod power operations on cohomology [6; Theorem

6.5].

Later, with similar techniques and a result
by T. Sugawara and H. Toda [45], J. Hubbuck computed
all possible types of a connected finite H-complex
without 2-torsion when the dimension of the generator
of H*(X;Z2) of highest degree is not of the form

23+1—1:

Lemma 2.4. [32; Theorem 1.2]. Let X be a
connected finite H-complex without 2-torsion. If the
dimension of the generator of H*(X;ZZ) of highest
degree is not of the form 28+1-1, then H*(X:ZZ)
is isomorphic as a Hopf algebra to H*(G;ZZ), where

G is one of the Lie groups U(n), SU(n+1),

S1 x Sp(n-l), or Sp(n), and n is the rank of X.

We remark here that in Lemma 2.3 and Lemma 2.4
it is originally assumed that H*(X;Q) is primitively
generated, i.e., there exists a set of generators which
are primitive, but this hypothesis is superfluous by a
result due to C. Curjel that X has a multiplication

such that H*(X;Q) is primitively generated; cf.[22:§8].

13

For a l-connected associative finite H—complex
(i.e., there exists a multiplication which is associative)
the types have also been determined for rank less than

or equal to 5 by various peOple listed below:

Lemma 2.5. Let X be a 1-connected associative
finite H-complex. If X is of rank one, then the type
of X is (3) [1 or 15]. If X is of rank 2, then
the type of X is (3,3),(3,5),(3,7), or (3,11) [40].
If X is of rank 3, then the type of X is
(3,3,3).(3,3,7),(3,5,7),(3.3.11), or (3.7.11) [37].

If X is of rank 4, then the type of X is
(3,3,3,3),(3,3.3,5).(3,3,5,5),(3,3,3,7).(3,3,5,7),
(3,3,7,7),(3,5,7,9).(3,3,3,11),(3,3,5,ll),(3,3,7,ll),
(3,7,7,ll),(3,3,11,11),(3,7,11,15), or (3,11,15,23)
[29 or 44]. If X is of rank 5, then the type

of X is (3,3,3,3,3),(3,3,3,3,5),(3,3,3,5,5).
(3,3,3,3,7),(3,3,3,5,7),(3,3,5,5,7),(3,3,3,7,7).
(3,3,5,7,7).(3,3,5,7,9),(3,5,5,7,9).(3,3,3,3,11),
(3,3,3,5,11),(3,3,3,7,11),(3,3,5,7,ll),(3,3,7,7,11),
(3,5,7,9,11),(3,3,3,ll,ll),(3,3,7,11,11),
(3,3,7,ll,15),(3,5,7,11,15),(3,7,9,11,15),(3,7,11,11,15).

(3,7,11,15,19), or (3,3,11,15,23) [25].

We remark here that the main technique used
in the proof above is the results by A. Clark in the

Paper "On #3 of finite dimensional H—spaces,"

14

appeared in Annals of Mathematics in 1963, and some

extensions of them.

63. On a relation between generators in H*(X;Zp) and

types of X.

Lemma 3.1. [13; Theorem 4.7]. Let X be a
path connected finite H-complex. If H*(X:Zp) has a
generator in dimension 2m-l, then H*(X;Q) has a
generator in dimension 2mpk-1, m > k 2_O. If
H*(X;Zp) has a generator in dimension 2m, then

H*(X:Q) has a generator in dimension 2mpk-l, m > k > 1.

Lemma 3.2. [13; Lemma 6.4]. Let X be a
connected finite H-complex. Let s be the smallest
integer for which HS(X) has p-torsion. Then
3 = 2n, and if p # 2, then HS(X;ZP) has a
primitive element. Further, if p = 2, and if n

is even, then HS(X:Z2) has a primitive element.

Lemma 3.3. [8 and 12]. Let X be a connected
finite H-complex. A necessary and sufficient condition
that X has no p-torsion is that its cohomology with
Z coefficients is an exterior algebra on odd dimensional

P
generators.

15

§4. Some facts about torsion in H-spaces.

The p-dimension of a space X is the largest

 

integer t such that Ht(X:Zp) # 0. Similarly, the

rational dimension of X is the largest t such that

 

Ht(X:Q) # 0.

Lemma 4.1. [12; Theorem 7.1]. Let X be a

 

path connected finite H-complex. Then for all p
the p-dimension of X equals the rational dimension

of X.

Lemma 4.2. [12: Corollary 7.2]. Let X be
as in Lemma 4.1 above. Then Ht(X) a Z and Ht_1(X)

is free.

For simply connected, path connected finite

H-complexes, we have:

Lemma 4.3. [12; Theorem 6.11]. Let X be a

 

l-connected finite H-complex. Then w1(X) = W2(X) = O.

The techniques used in §3 and §4 above are a
close study of Hopf algebra structure, Bockstein
spectral sequence, biprimitive spectral sequence,
Serre spectral sequence, and Leray-Cartan spectral
sequence (on the covering spaces of H—spaces). The
reader is referred to the well-written papers quoted

above and the paper by J. Milnor and J. Moore [33].

16

55. On a relation between the homology of X and the

homology of 0X.

Even when X is a finite H-complex, fix will not
be a finite H-complex unless X has the homotopy type
of K(G,2), where K(G,2) is the Eilenberg-MacLane
complex and G is finite free abelian [17]. A

theorem by W. Browden states that:

Lemma 5.1. [13: Theorem 5.15]. Let X be

 

a path connected, simply connected H-space. Suppose
H*(X:K) = A(xl,...,xm,...), an exterior algebra on

generators xl,x ,..., dim xi = 2ni + 1, K a field.

2
Then H*(QX:K) = K[yl....,ym,...], a polynomial
algebra on generators yl,...,ym,... with

dim yi = 2ni.

We remark here that the lemma above is true
even without the hypothesis that X is an H—space.
The proof can be shown by using Serre's theory of
classes of abelian groups or by using the spectral

sequence of the cobar construction.

Lemma 5.2. [20; Theorem 4.1]. Let X be

 

a path connected, simply connected H—space of finite
homological type and let si be the homology

mod p suspension in degree i:

17

Si : Q(Hi(0X:Zp)) 4 P(Hi+1(X:Zp))
from the subspace of indecomposable elements of
Hi(flX:Zp) to the subspace of primitive elements of

Hi+l(X;Zp). Then if p=2, si 18 a monomorphism
i

unless = 2q(2km+2) - 2 for q > O, k > O, and
Q(Hm(flX;Z2)) # O and si is an epimorphism unless

i = 2km+l for k > O and Q(Hm(QX;Z2)) # 0.

We remark here that the above result is a
slight improvement over that of [13; Theorem 5.13].
The technique of proof is an application of Eilenberg—

Moore spectral sequence.

Lemma 5.3. [19]. If X is 1—connected finite

 

H—complex and if fix is torsion free, then X has

no higher torsion.

Lemma 5.4. [13; Theorem 6.6]. Let X be a

 

path connected finite H—complex. Suppose that 0X has
no torsion. If s is the smallest integer for which

HS(X:Z) has p—torsion, then s = 2n, and n a 1 mod p.

§6. On localization of H-complexes.

One of the powerful techniques in attacking

finite H-complex problems is the concept of localization

18

of H-complexes. Localization of CW complexes has been
studied by several mathematicians, notably by J. Adams
[2], D.W. Anderson [5], Bonsfield-Kan [10], Curtis-
Mislin [23], Mimura-Nishida-Toad [35], D. Sullivan
[46], and others. A relatively complete list of
references can be found in [35]. For a description

of the construction of X(p), the localization of

X at the prime p, see one of the above references.

The principal results on X(p) are listed below:

Lemma 6.1. [35: Theorem 2.4]. The correspondence

 

X 4 X is a functor from the homotopy category of

(p)

1—connected CW complexes of finite type to the

homotOpy category of l-connected countable CW complexes.

Lemma 6.2. [35; Theorem 2.5]. Let X be a
l-connected CW complex of finite type. Then
H*(X(p)) a H*(X) Q>Q(p), where Q(p) denotes the
integers localized at p, (i.e., the ring of rationals
whose denominators when reduced to the simplest form

are prime to p.)

Lemma 6.3. [35; Theorem 7.1]. Let X be an

H—space. Then X(p) is also an H-space.

19

Lemma 6.4. [35; Proposition 2.2]. Let X be

 

a 2—connected CW complex of finite type. Then

(OX) has the homotopy type of Q(X(p)).

(p)

§7. The cohomology of covering spaces of H-spaces.

Let X be a connected finite H-complex. Then
in [39] it is shown that X, the universal covering
space of X, is a l-connected finite H—complex. Let
X 'be a covering space of X. If X E.x is the
covering projection, then we can convert X‘E X to

g K(G,l), where i

the fibration sequence X 5.X'
is the inclusion map and X' has the homotopy type
of X and consider the Leray-Cartan spectral

sequence of this fibration sequence. We have:

Lemma 7.1. [11]. Let p be an odd prime.

 

Then H*(X:Z) a A ® E as rings, where A = w*(H*(X:Zp))
aiH*(X:Zp)/I, I is the ideal generated by
f*(H*(K(G,l):Zp)) and E is the exterior algebra

on n generators X1"°"Xn , where the dimension
ri r1 rn
of xi is 2p -1, and 2p ,...,2p are the

dimensions of a system of generators of the kernel
of f*. If p=2, then the same result holds, but

the isomorphism is only as modules.

20

Lemma 7.2. [11]. As in Lemma 7.1 above if

 

N

X = X, then the ideal I is generated by H1(X;Zp)
and the subspace B of H2(X:Zp), where each element
of B lies in the image of the Bockstein homomorphism

8 : H1(X;Zp) 4 H2(X;Zp) for every 5. The dimension

of the xi's is determined by writing the algebra
generated by B as the tensor product of polynomial
rings and truncated polynomial rings. FOr each
truncated polynomial ring on one generator we get one
xi whose dimension is the height of the generator

minus one.

§8. Statements of Results.

Throughout this section, X denotes a path
connected finite H-complex and OX denotes the loop
space of X. Also assume that if X is a l-connected

finite H-complex of rank 2 having 2-torsion, then
* o a. * 0

where 62 is the exceptional Lie group whose cohomology

with Z2 coefficients is:

H*(G2:Z2) a zz[x3]/(x§) ® A (Sq2x3).

21

the tensor product of a polynomial algebra on one
generator of dimension 3 truncated at height 4 and

an exterior algebra on one generator of dimension 5.

We remark here that this hypothesis might
be superfluous as the proof of it was claimed [unpublished]
by J.R. Hubbuck during the International Conference

on H—spaces at Neuchatel in Switzerland in 1970.

As mentioned in Chapter one it has been shown
by W. Browder that if X is of rank one, then X

has no odd torsion. If X is of rank 2, we have:

Theorem 1. If X has rank 2, then X has

p—torsion for p 2.5.

Theorem 2. If X is l—connected and has rank

2, then X has no odd torsion.

Torsion in X is closely related to torsion

in 0X. A relation of them is the fOllowing:

Lemma 3. Let p be a prime. If X has no

p-torsion, then 0X has no p-torsion.

For most known examples of 1-connected finite
H-complex X, ox has been shown to be torsion free:
thus it is a long standing conjecture that OX is

torsion free. In required to this we have:

22

Theorem 4. If X has rank 2, then 0X

 

is torsion free.

Recall that if H1(X;Z) has an element of order
p2 for some i and some prime p, then we say that

X has higher p-torsion.

Corollary 5. If X is l-connected and has

 

rank 2, then X has no higher p-torsion for any p.

If X has rank 2, then from Theorem 1 above
we have that X has no p-torsion for p 2,5. In
the absence of 2-trosion we have an interesting

restriction on the presence of 3-torsion.

Corollary 6. If X has rank 2 and has

 

no 2-torsion, then X has 3-torsion if and only if

H2(X;Z) has an element of order 3.

This completes the study of torsion in the

rank 2 case. For ranks higher than 2 we have:

Theorem 7. (i) If X has rank less than or

 

equal to 5 and if X has
no 2-torsion, then X has
no p-torsion for p 2_7.
(ii) If X is l-connected, has
rank less than or equal to
4, and has no 2-torsion, then

X has no p-torsion for p.2 5.

23

The center of SU(n) is isomorphic to Zn .
a cyclic group of order n [3]. Thus for any prime
p the corresponding projecture group SU(p)/Zp has
p-torsion. However, these groups are not simply
connected. For l—connected finite H-complexes having
no 2-torsion we have Theorem 7 above. For l-connected

associative finite H—complexes we have:

Theorem 8. Let X be a l-connected associative
finite H-complex of rank less than or equal to 5.
Then X has no p-torsion for p 2_7. Further, OX

has no p-torsion for p 2_7.

For a connected topological group or loop space
G with H*(G;Zp) finitely generated, if p is an odd
prime and if H*(G:Z) has p-torsion, then H*(G:Zp)
is not primitively generated. This is a result by W.
Browder [16; Theorem 1]. A recent result of J.
Hubbuck [31; Corollary 1.3] states that if Y is a
l-connected homotopy commutative and homotopy associative
H—space and if H*(Y;Z) has no p—torsion where p is
an odd prime, then H*(Y;Zp) is primitively generated
if and only if the ring H*(Y:Zp) is isomorphic to

Zp[y1,...,ym,...] ® A(x1,...,xn,...), a tensor product

24

of a polynomial algebra on generators yi all having
dimension 2 and an exterior algebra on odd dimensional

generators xj.

As a further application of the techniques of
this thesis, we give a simple proof of Hubbuck's
result quoted above for a l-connected finite H—complex
X ‘when X has rank less than or equal to 4 and

has no 2-torsion.

CHAPTER I II

Proofs of Results

Throughout this chapter X denotes a path
connected finite H-complex and OX denotes the 100p
space of X. We also assume that if X is l-connected

finite H-complex of rank 2, and if X has 2-torsion,
* . a: * o
H (X.Z2) — H (G2'22)'

where the cohomology of G is stated in §8 of

2
Chapter II. Again we remark here that this hypothesis
might be superfluous as the proof of it was claimed

[unpublished] by J.R. Hubbuck during the International

Conference on H-spaces at Heuchatel in Switzerland

in 1970.

Theorem 1. If X has rank 2, then X has

 

no p-torsion for p‘Z S.

nggf, We divide into two cases, namely,
(i) X has no 2-torsion, and (ii) X has 2-torsion.
(i) If X has no 2-torsion, then the type
of X is (l,l).(1,3).(l,7).(3,3),(3,5),

(3,7), or (7,7) by Lemma 2.2 in Chapter II.

25

(ii)

26

Suppose that X has p-torsion for
p 2.5. If s is the smallest
integer for which HS(X;Z) has
p—torsion. then by Lemma 3.2 in
Chapter II, s=2m and, since

p # 2, HS(X;Zp) has a primitive
element. From Lemma 1.1 in

Chapter II we have P(H2m(X:Zp)) E
(Q(H2m(X;Zp)))*. since H*(X;Zp)
and H*(X;Zp) are dual to each
other. Thus H2m(X:Zp) has an
indecomposable element: hence a
generator. By Lemma 3.1 in

Chapter II H*(X:Q) has a generator
in dimension 2mpk-l, for some

k, 0 < k < m. We have 2m 2’2.

p 2_5. and k 2_l: so 2mpk—l'2 9.
This contradicts the possible types
of X which are (1,1),(1,3),(l,7).
(3,3). (3.5). (3,7). and (7.7).

If X has 2-torsion, then again we
consider two cases, (a) X is
simply connected, and (b) X is

not simply connected.

27

(a) If X has 2-torsion and is simply connected,
then H*(X:Z2) a H*(G2;Z2). where the exceptional Lie

group G2 has mod 2 cohomology:

H*(GZ:Z2) a Z2[x3]/(x§) ® A(S:x3).
the tensor product of a polynomial algebra on one generator
of dimension 3 truncated at height 4 and an exterior
algebra on one generator of dimension 5. Since
H14(G2:ZZ) # O and Hi(G2:ZZ) = O for i > 14, we
have H14(X;Z2) #'O and Hi(X;Z2) = O for i > 14.
By Lemma 4.1 in Chapter II we have Hi(X7Q) # O and
Hi(X;Q) = O for i > 14. Now, if X has p-torsion
for p'2 5, then by the argument used in part (i)
above we see that there is a generator for H*(X:Q)
in dimension 2mpk-l, where 2m is the smallest
integer for which H2m(X;Z) has p-torsion and O < k g m.
Since X is simply connected, we have W1(X) = w2(X) = O
by Lemma 4.3 in Chapter II. This implies that
Hi(X:Z) = H2(X:Z) = 0. Thus. 2m 2_4, p‘Z 5. k 2.1,
and so 2mpk-l 2.19. This contradicts the fact that
Hi(X:Q) = O for i > 14.

(b) If X has 2-torsion and is not simply
connected, then consider the universal covering space

X of X. First, observe that X does not have type

(1.1). If X has type (1.1). then H2(X;Z) a Z

28

and Hi(X:Z) = O for i > 2 by Lemma 4.2 in Chapter II.
Since X has 2-torsion, H2n(X;Z) contains an element
of order 2r(r 2_l), where 2n is the smallest integer
for which H2n(X;Z) has 2~torsion. Since 2n 3 2.

we have a contradiction; hence X does not have type
(1.1). Also notice that X is simply connected finite
H-complex [see §7 in Chapter II]. From Lemma 7.2 in
Chapter II we have that if (n1.....ni) is the type

of X. then the possible types of X are (n1....,ni)
and (l.....l,nl.....ni). Let X have type (n1).

i.e., X has rank 1. Then by Lemma 2.1 in Chapter II
we have nl = 3 or 7. This implies that X has type
(1.3) or (1.7), in which case there is no p-torsion
for p 2.5 by the same argument used in (i) above.

Now let X have rank 2. From Lemma 2.2 in Chapter

II we see that if X, has no 2-torsion. then X
has type (3,3),(3,5),(3,7), or (7.7). So the type
of X is (3,3),(3,5),(3,7), or (7,7). As
reasoned in (i) above, X has no p-torsion for

p 2_5 when X has no 2-torsion. If X has 2—

torsion. then again we have:
* . =5- * o
H (x.zz) ... H (62.22).

X has no p-torsion for p 2_7 because if it does

H*(X;Q) will have a generator in a dimension at least

29

2(l)(7) - l = 13. This implies that the type of X

is (1.13) since the t0p dimension of H*(X;Q) is

14. But then X, is of rank one. a contradiction to
our hypothesis that X is of rank 2. Suppose X has
S-torsion. Then H*(X:Q) has a generator in dimension
2m5k-l for m.2 l. k 2_l. where 2m is the

smallest integer for which H2m(X;Z) has S-torsion.
The only dimension to consider is 9 since the top
dimension is 14. Again this implies that the type

of X is (5.9). Now, X has 2-torsion, so let

2n be the smallest dimension in which H*(X;Z) has
2-torsion. We see that 2n is not 2.4, or 8 since
then H*(X7Q) will have a generator in dimension
2(1)2k-l, 2(2)2k-1, or 2(4)2k-l, respectively,
which is not 5 or 9. Thus 6 and 10 are the

only remaining possibilities. In either case

H3(X7ZZ) = H4(X:ZZ) = 0. But H (£22) 2H3(G2;Z2) a- 22

so by Lemma 7.2 in Chapter II. we have a contradiction.

This completes the proof of the theorem.

Remark. The theorem above says that X has
no p-torsion for p 2.5. In fact. X may have 2-

torsion: simply let X = G Also, X may have 3-

2.
torsion. Consider SU(3). Its center is 23 [3].

Let X be the corresponding projective group PSU(3).

It is Obvious that X is of rank 2 and H2(X:Z) a 23.

30

Theorem 2. If X has rank 2 and is l-connected.

then X has no odd torsion.

grggf, We divide into two cases, namely,
(i) X has no 2-torsion, and (ii) X has 2-torsion.
(i) Let X have no 2-torsion. Since X
is simply connected, we have w1(X) =
W2(X) = O by Lemma 4.3 in Chapter II.
This implies that H1(X) = 32(x) = 0;
hence the type of X is (3,3),(3,S),
(3.7), or (7.7), by Lemma 2.2 in
Chapter II. Suppose that X has p-
torsion for p.2 3. Let s be the
smallest integer for which HS(X;Z)
has 3-torsion. Then by Lemma 3.2 in
Chapter II. s=2m and HS(X:ZP)
has a primitive element. From Lemma
1.1 in Chapter II we have that
P(H2m(X:Zp)) 2 (Q(H2m(X:Zp)))*. since
H*(X:Zp) and H*(X:Zp) are dual
to each other. Thus H2m(X:Zp) has
an indecomposable element: hence
a generator. By Lemma 3.1 in
Chapter II, H*(X:Q) has a generator
in dimension 2mpk-l, for some k,

0 < k < m. We have that 2m 2.4

(ii)

31

since X is simply connected, p‘Z 3,
and k 2 1; so 2mpk-l 2 4-3-1 = 11.
This contradicts the possible types of
X which are (3,3),(3,5),(3,7), and (7.7).
Suppose that X has 2—torsion. From
Theorem 1 X‘ has no p—torsion for

p 2_5. Thus all we need to show is
that X has no 3-torsion. Suppose
the contrary holds. Then X has
2-torsion and 3-torsion. Consider
X(3). the localization of X at

the prime 3. By Lemma 6.1 in
Chapter II. X(3) is a l-connected
CW complex and by Lemma 6.3 in
Chapter II we see that X(3) is a
l-connected finite H-complex.
Since X is of rank 2, X(3) is
of rank 2 by Lemma 6.2 in Chapter
II. Also by Lemma 6.2 in Chapter II

we have that X has only 3-torsion

(3)
which implies that X(3) has no
2-torsion. Now, by the result in
part (i) above we see that this is a
contradiction. Thus X has no p-

torsion for p 2.3 and this completes

the proof of the theorem.

32

Lemma 3. Let p be a prime and X be
l—connected. If X has no p-torsion, then OX

has no p-torsion.

Proof. Suppose that OX has p-torsion for
some prime p. Then by the Universal Coefficient

Theorem for homology:
Hn(OX:Zp) 2 (Hn(OX:Z) ® Zp) m (Hn_l(OX;Z)*Zp)

we have that if Hn(OX;Z) has p-torsion, then
Hn(OX:Zp) # O and Hn+l(OX:Zp) #'O. This means that
Hi(OX:Zp) # O for some positive odd integer i. This
implies that H*(OX:Zp) # Zp[y1,...,ym....], where

° = . * .
dim yi 2ni. hence H (X.Zp) # A(xl,....xm....), an
exterior algebra on odd dimensional generators by
Lemma 5.1 in Chapter II. But then X has p-torsion
by Lemma 3.3 in Chapter II. This contradicts the
hypothesis that X has no p-torsion. Thus OX

has no p—torsion.

Theorem 4. If X has rank 2, then OX

has no p-torsion for any p.

Proof. We divide into 2 cases, (i) X is
simply connected, and (ii) X is not simply connected.
(i) Suppose that X is simply connected.

If X has rank one, then X has the

homotOpy type of 83 or S7 [15:

Theorem 5.2] and 083 and OS7 are

33

torsion free. Suppose now that X
has rank 2. By Theorem 2 X has no
p-torsion for p‘Z 3. Thus by Theorem
3 above OX has no p-torsion for
p.2 3. So, if we can show that OX
has no 2-torsion, then we are done.

If X has no 2-torsion, then
by Lemma 3 OX has no 2-torsion and
the theorem is proved. Suppose that
X has 2-torsion. Since X is 1-
connected,

* o a: * o

where the cohomology ring H*(G2:Z2)
has one generator in dimension 3

and one generator in dimension 5.

We shall show that OX has no 2-
torsion. Suppose the contrary holds.
As reasoned in the proof of Lemma 3,
Hi(OX:Zz) contains 2-torsion for some
positive odd integer i. Let m

be the smallest such i. Then
Hm(OX:Z2) contains an indecomposable
element. Since. by Lemma 5.2 in

Chapter II,

Sm : Q(Hm(OX:zz)) 4 P(Hm+1(x;zz))

(ii)

34

is a monomorphism for m # 2t(2n)—2,
where t.2 1 and 2n = dimension
of some generator of H*(X:ZZ). we

see that H

m+1(X;Z2) contains a

primitive element. Since

P(H (X;ZZ)) e.- (Q(Hm+1(X7Z2)))*.

m+1
we see that Hm+1(X;Z2) has an
indecomposable element: hence a
generator. But m+1 is even. a
contradiction to the fact that
H*(X:ZZ) has generators only in
dimensions 3 and 5.

If X is not simply connected,

then consider the universal covering
space X' of X. Let (OX)*

denote the path connected component
of OX containing the base

point * and let p : X 41X be

the covering projection. Since

0X is path connected and (OX)*

is also path connected, the map

Op : OX,» (OX)* induces a one-

one correspondence between the

35

path component of OX and that of
(OX)*. Since p# :W1(X) swim)
for i 2'2 and wi(Y) a ni_l(OY)
for any Y, we have that

mp)#: wi_1m§) EETTj__1(QX) -_- wi_1((OX)*).
Thus OX has the weak homotopy

type of (OX)* and thus OX 'has

the homotopy type of (OX)* .

cf. [41: Chapter 7]. But X, is

a simply connected finite H-complex

of rank 1 or rank 2. so by

part (i) above OX, is torsion free.
Thus (OX)* is torsion free. The
cohomology of OX is the direct sum

of the cohomology of the path components
of OX. Since all path components of

OX have the homotopy type of (OX)* .

it follows that OX is torsion free.

Corollary 5. If X is l-connected and is of
rank 2, then X has no higher p-torsion for any

prime p.

Proof. By Theorem 4 above OX is torsion free.
By Lemma 5.3 in Chapter 2. X has no higher p-torsion

for any prime p.

36

Corollary 6. If X is of rank 2 and has
no 2-torsion. then X has 3-torsion if and only if

H2(X;Z) has an element of order 3.

Egggf, If X has no 2-torsion, then the
type of x is (1.1).(1,3).(1.7).(3.3).(3.5).(3.7).
or (7.7). Suppose X has 3-torsion. Then H*(X:Q)
has a generator in dimension 2m3k-l, where
O < k < m and 2m is the smallest integer for
which H2m(X;Z) has 3-torsion. But 2m cannot
be greater than or equal to 4 because if 2mI2 4,
then we have 2m-3k-lI2 11, a contradiction. Thus

2m = 2.

Remark. As in the remark following the proof
of Theorem 1, PSU(3) has no 2-torsion but has 3-

torsion and H2(PSU(3):Z) a 23.

Theorem 7. (i) If X has rank less than

 

or equal to 5 and if X has no 2-torsion, then X
has no p-torsion for p.2 7.

(ii) If X is l—connected, has
rank less than 5, and has no 2-torsion, then X

has no p-torsion for p‘2 5.

37

Proof.

(i)

Let X be of rank less than or equal
to 5 and have no 2-torsion. Then

by Lemma 2.3 in Chapter II, the type
of X is a union of sets taken from
(l).(3).(7).(3.5).(3.7).(3.5.7).
(3,7,11),(3,5.7.9).(3,7,11,15).
(3,5,7,9,ll). or (3,7,11,15,19).
Suppose X has p-torsion for some

p‘Z 7. If s is the smallest integer
for which HS(X;Z) has p-torsion, then
by Lemma 3.2 in Chapter II, s = 2m
and. since p # 2, then HS(X;ZP)

has a primitive element. From

Lemma 1.1 in Chapter II we have that
P(H2m(X;Zp)) a (Q(H2m(X;Zp)))*, since
H*(X;Zp) and H*(X:Zp) are dual to
each other. Thus H2m(X;Zp) has an
indecomposable element: hence, a
generator. By Lemma 3.1 in Chapter II.
H*(X:Q) has a generator in dimension
2mpk_1. for some k, 0 < k < a. Let
p=7. If 2m=2 and k=l, then

2mpk-1 = 2-7-1 = 13, a contradiction

to the possible types of X. If

(ii)

38
2m=2, k'Z 2, then 2mpk-l 2_2°72—1 = 97,
again a contradiction. If 2m=4 and
k 2_1, then 2mpk-l 2_4-7-1 = 27. Thus
in any case X has no 7—torsion. Let
p‘Z 11. If 2mIZ 2 and k'Z 1, then
2mpk-l 2.2-11-1 = 21, which is a
contradiction to the possible types of
X. Thus X has no p-torsion for p'Z 7.
Let X be of rank less than or equal to
4 and have no 2-torsion. Then the type
of X is a union of sets taken from
(1).(3).(7).(3.5).(3.7).(3.5.7).(3.7.11).
(3.5.7.9), or (3,7,11,15). By the same
argument used in (i) above, we have
that H*(X:Q) has a generator in
dimension 2mpk-l, where O < k < m
and 2m is the smallest integer for
which H2m(X:z) has p-torsion for
some p‘g 5. Since X is l-connected,
W1(X) = w2(X) = O by Lemma 4.3 in
Chapter II; hence H1(X) = H2(X) = 0.
Thus 2m 2_4. k‘Z 1, and p‘Z 5: so
2mpk—l 2 4.5-1 = 19. This contradicts

the possible types of X.

39

Remark (i). If X is a connected finite
H—complex and if X is also homotOpy commutative. then
X has no 2-torsion [14: Corollary 8.7]. If X is a
connected finite H-complex both homotopy commutative and
homotopy associative. then X is torsion free [14:
Theorem 8.10]. If G is a compact Lie group with
trivial center, it follows that G does not admit any

homotopy commutative and homotopy associative multiplication.

Remark (ii). For all the known compact connected
Lie groups G we have that if G has torsion, then
it has 2-torsion. On the other hand. there are finite
H—complexes that have 3-torsion or 5-torsion but not
2-torsion [35: § 8]; however, their ranks are not less

than or equal to 5.

Theorem 8. If X is l-connected. associative,
and has rank less than or equal to 5, then X has no
p-torsion for p 2.7: furthermore, OX has no p-torsion

for p]; 7.

nggf, Let X be a l-connected, associative,
finite H-complex of rank less than or equal to 5. Then
by Lemma 2.5 in Chapter II the type of X is (3),(3.3).
(3.5)o(3.7).(3.11).(3:3.3).(3.3.5).(3a3.7).(3:5.7).(3.3.11).

(3.7.11),(3,3.3,3),(3,3,3,5),(3.3.5.5),(3,3,3,7),(3,3,5,7),

4O

(3.3.7.7).(3.5.7.9).(3.3.3.11).(3.3.5.11).(3.3.7.11),
(3.7.7.11).(3,3,11,11).(3,7.ll.15),(3,11,15,23),(3.3.3.3.3).
(3.3.3.3.5),(3.3.3.5.5).(3.3.3,3.7),(3,3,3,5.7).(3.3.5.5.7).
(3.3.3.7.7),(3.3.5.7.7),(3.3.5.7,9),(3,5.5.7,9).(3.3.3.3.11),
(3.3.3.5.11).(3.3.3.7.11).(3.3.5.7,11).(3.3.7.7.11).
(3,5,7,9,ll).(3.3,3.ll,ll).(3.3.7.11,1l),(3.3.7.11,15).
(3.5.7,1l,15),(3.7.9.11,15),(3,7,11,11,15),(3,7,11,15,19),
or (3,3,11,15,23). By using the same argument as in (i)

of Theorem 7 above we see that if X has p-torsion for

pig 7 and if 2m is the smallest integer for which
H2m(X;Z) has p-torsion, then H*(X;Q) has a generator

in dimension 2mpk-l for some k, 0 < k < m. Let

p=7. If 2m=2, k=l, then 2mpk—1 = 2-7-1 = 13, a
contradiction to the possible types of X. If 2m=2,

k‘z 2, then 2mpk-l 2_2-72-l = 97, again a contradiction.
If 2m=4 and k 2_l, then 2mpk-l 2_4°7-l = 27. Thus

in any case X has no 7—torsion. Let p=ll. If

2m=2, k=l, then 2mpk-l = 2°ll-l = 21. a contradiction.
If 2m'2 2, k 2,2, then 2mpk—l‘2 2-ll—l = 241. again

a contradiction. Let p 2.13. If 2m 2_l. k'g 1,

then 2mpk-l‘2 2-13-1 = 25. Thus X has no p—torsion

for p 2.7. The last statement of the theorem follows

from Lemma 2.

41

Remark. Let X be l-connected, have rank less
than or equal to 4, and have no 2-torsion. If all
generators in H*(X:Zp) for p 2.5 have dimension 3.
then H*(OX;ZP) is primitively generated. This is a

special case of Hubbuck's result [31: Corollary 1.3].

Proof. If X is l—connected, has rank g_4.
and has no 2-torsion, then by (ii) of Theorem 7 above X
has no p-torsion for p 2’5. Therefore by Lemma 3.3

in Chapter II
H*(X;Zp) =_-A(xl.....xi). 13134. p25,

an exterior algebra on odd dimensional generators. By
hypothesis dim xj = 3 for l g_j g_i. It follows that

H*(OX:Z ) e z [yl,...,yi], 1 g i 34, p 2 5 and that

P P
dim yj = 2 for l g_j g_i by Lemma 5.1 in Chapter II.
Since X is l-connected, X is 2-connected by Lemma

4.3 in Chapter II. Thus OX is l-connected because
w1(OX) = W2(X) = 0. It follows that H1(OX:Zp) = 0.
Now. dim yj = 2 for l g_j g_i and H1(OX:ZP) = 0;
so if A* is the homomorphism induced by the diagonal
map A : OX 4 OX x OX, then

. = . 1 + 1 R) . H. X72 H. 7Z
A*(yj) yJ so .yJ ei+2j3=2 1m p) s J(ax p).
i.e., yj is primitive. Therefore H*(OX;Zp) is

primitively generated.‘

[1]

[2]

Bibliography

J.F. Adams. "On the Non-existence of Elements of
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