Self-homeomorphisms and Degree $$\pm \,1$$ Self-maps on Lens Spaces

Abstract

In this article, we give a sufficient and necessary condition to that a degree $$\pm \,1$$ self-map on a $$(2n-1)$$ -dimensional lens space for $$n \ge 2$$ is homotopic to a self-homeomorphism. In fact, we show how the endomorphism induced by a homeomorphism can act on the fundamental group of the lens space. Moreover, we provide a specific description of a class of lens spaces which admit self-homeomorphisms inducing nontrivial automorphisms on the fundamental groups. Furthermore, the topologically conjugate classification for a special class of periodic homeomorphisms on $$S^{2n-1}$$ is obtained.