FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 23

Then we have the following result :

7.5 Proposition Let

X

=

sml

X ... X

smp

X

snl

X ... X

snq'

with

mj

1 odd

and

nk

even. Then £*#(X) is infinite if and only

if

condition 7.4 is satisfied with

m

=

(m1, ... , mp)

and

n

=

(n1, ... , nq).

Proof Suppose condition 7.4 is not satisfied.

If[/]

E

£#

(M)

with

!Iva

=

Llvo,

then

in the expression (7.1) for

f,

each

.\~,'7

=

0. Hence 01(f,L)

=

0 and so

f

~

L

by

Proposition 7.3. Thus £#(M)

= 1.

By Remark 5.9, £*#(X) is finite.

Conversely, if7.4 is satisfied, then for each)..

E

IQ

we define a map f.. :

M ----. M

by f.. (

Wj)

=

Wj

+

)..uEv'J

and f.. is the identity on all other generators. For two

such maps

f..

and /

11

,

we have

Therefore

{[f.,]

I ).. E

IQ}

is an infinite subset of £#(M) by Proposition 7.3. By

Remark 5.9, £*#(X) is infinite. D

Although the hypotheses of Proposition 7.5 appear cumbersome, they are often

easy to verify in practice.

If

all the spheres are odd dimensional (q

=

0) or if

all the spheres are even dimensional

(p

=

0), then £*#(X) is finite. However, a

product of only three spheres can yield

£*#(X) infinite. For example, let X

S2k+l x S2k+4 x S2k+6. Then, for each k

=

1,2, ... , £*#(X) is infinite.

References

[Ar]

Arkowitz, M., Formal Differential Graded Algebras and Homomorphimsms,

J.

of Pure and Appl. Alg.,

51

(1988), 35-52.

(A-C] Arkowitz, M. and Curjel, C., Groups of Homotopy Classes, Springer Lecture

Notes

in

Math.,

4

(1964).

[Ar-Lu)

Arkowitz, M. and Lupton, G., On the Nilpotency of Subgroups of Self-Homo-

topy Equivalences, in preparation.

[Au-Le)

Aubry, M. and Lemaire, J.-M., Sur Certaines Equivalences d'Homotopies, Ann.

Inst. Fourier, 41 (1991), 173-187.

[D-G-M-S) Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real Homotopy Theory

of Kahler Manifolds, Invent. Math.,

29

(1975), 245-274.

[Di)

Didierjean, G., Homotopie de l'Espace de Equivalences d'Homotopie Fibrees,

Ann. Inst. Fourier, 35 (1985), 33-47.

Then we have the following result :

7.5 Proposition Let

X

=

sml

X ... X

smp

X

snl

X ... X

snq'

with

mj

1 odd

and

nk

even. Then £*#(X) is infinite if and only

if

condition 7.4 is satisfied with

m

=

(m1, ... , mp)

and

n

=

(n1, ... , nq).

Proof Suppose condition 7.4 is not satisfied.

If[/]

E

£#

(M)

with

!Iva

=

Llvo,

then

in the expression (7.1) for

f,

each

.\~,'7

=

0. Hence 01(f,L)

=

0 and so

f

~

L

by

Proposition 7.3. Thus £#(M)

= 1.

By Remark 5.9, £*#(X) is finite.

Conversely, if7.4 is satisfied, then for each)..

E

IQ

we define a map f.. :

M ----. M

by f.. (

Wj)

=

Wj

+

)..uEv'J

and f.. is the identity on all other generators. For two

such maps

f..

and /

11

,

we have

Therefore

{[f.,]

I ).. E

IQ}

is an infinite subset of £#(M) by Proposition 7.3. By

Remark 5.9, £*#(X) is infinite. D

Although the hypotheses of Proposition 7.5 appear cumbersome, they are often

easy to verify in practice.

If

all the spheres are odd dimensional (q

=

0) or if

all the spheres are even dimensional

(p

=

0), then £*#(X) is finite. However, a

product of only three spheres can yield

£*#(X) infinite. For example, let X

S2k+l x S2k+4 x S2k+6. Then, for each k

=

1,2, ... , £*#(X) is infinite.

References

[Ar]

Arkowitz, M., Formal Differential Graded Algebras and Homomorphimsms,

J.

of Pure and Appl. Alg.,

51

(1988), 35-52.

(A-C] Arkowitz, M. and Curjel, C., Groups of Homotopy Classes, Springer Lecture

Notes

in

Math.,

4

(1964).

[Ar-Lu)

Arkowitz, M. and Lupton, G., On the Nilpotency of Subgroups of Self-Homo-

topy Equivalences, in preparation.

[Au-Le)

Aubry, M. and Lemaire, J.-M., Sur Certaines Equivalences d'Homotopies, Ann.

Inst. Fourier, 41 (1991), 173-187.

[D-G-M-S) Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real Homotopy Theory

of Kahler Manifolds, Invent. Math.,

29

(1975), 245-274.

[Di)

Didierjean, G., Homotopie de l'Espace de Equivalences d'Homotopie Fibrees,

Ann. Inst. Fourier, 35 (1985), 33-47.