Abstract
This paper observes that the induced homomorphisms on cohomology groups by a cyclic map are trivial. For a CW-complex X, we use the fact to obtain some conditions of X so that the n-th Gottlieb group is trivial for an even positive integer n. As corollaries, for any positive integer m, we obtain which are due to D. H. Gottlieb and G. Lang respectively, where is the 2m- dimensional sphere and is the complex projective m-space. Moreover, we show that is the quaternionic projective m-space for any positive integer m and II is the Cayley projective space.
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