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.
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