On the connection of periodic homeomorphisms of a surface with Seifert manifolds and the Morse-Smale diffeomorphism

In this paper we consider periodic homeomorphism $\varphi$ which acts on genus $p$ surface. Homeomorphism is called periodic if exists $n\in \mathbb{N}$ such that $\varphi^{n} \equiv \mathrm{id}$. We study connections of such homeomorphisms with 3-dimensional topology. More accurately, we have established the condition that given 3-dimensional Seifert manifold is realised as mapping torus of some periodic homeomorphism $\varphi$. Moreover, this periodic homeomorphism is almost fully determined by topology of its mapping torus. This connection allowed us to proof, for instance, that there are no homotopy identical periodic homeomorphisms without points of smaller period on surfaces of positive genus. Using the connection between Morse-Smale diffeomorphisms and periodic homeomorphisms, we succeeded in classification of corresponding periodic homeomorphism of arbitrary Morse-Smale diffeomorphism with one source, one sink and one saddle orbit with negative type of orientation, what can be used in solution of problem of realisation of arbitrary Morse-Smale diffeomorphism on 2-dimensional manifold.