A remark on the genus of the infinite quaternionic projective space

Abstract

It is shown that only countably many spaces in the genus of $\hpinfty$, the infinite quaternionic projective space, can admit essential maps from $\cpinfty$, the infinite complex projective space. Examples of countably many homotopically distinct spaces in the genus of $\hpinfty$ which admit essential maps from $\cpinfty$ are constructed. These results strengthen a theorem of McGibbon and Rector which states that among the uncountably many homotopy types in its genus, $\hpinfty$ is the only one which admits a maximal torus.