Cohomology and trivial Gottlieb groups

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 $G_{n}(X)$ is trivial for an even positive integer $n$. As corollaries, for any positive integer $m$, we obtain $G_{2m}(S^{2m})=0$ and $G_2(CP^m)=0$ which are due to D. H. Gottlieb and G. Lang respectively, where $S^{2m}$ is the $2m$- dimensional sphere and $CP^m$ is the complex projective $m$-space. Moreover, we show that $G_4(HP^m)=0$ and $G_8(\Pi)=0$, where $HP^m$ is the quaternionic projective $m$-space for any positive integer $m$ and $\Pi$ is the Cayley projective space.