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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