Periods of continuous maps on some compact spaces

Abstract

The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: ${\mathbb{S}}^n$ (the n-dimensional sphere), ${\mathbb{S}}^n\times {\mathbb{S}}^m$ (the product space of the n-dimensional with the m-dimensional spheres), $\mathbb{C}{P}^n$ (the n-dimensional complex projective space) and $\mathbb{H}{P}^n$ (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

Keywords:

• Periods
• periodic points
• continuous map
• Lefschetz fixed point theory
• sphere
• product of two spheres
• complex projective space
• quaternion projective space

AMS Subject Classifications:

• 37C25
• 37C30

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author of this work was partially supported by MINECO [grant number MTM2014-51891-P]; Fundación Séneca de la Región de Murcia [grant number 19219/PI/14]. The second author is partially supported by the MINECO [grant number MTM2016-77278-P] and [grant number MTM2013-40998-P]; an AGAUR [grant number 2014SGR-568], [grant number FP7-PEOPLE-2012-IRSES 318999].