A periodicity criterion and the section problem on the Mapping Class Group

TL;DR

This paper introduces a periodicity criterion for surface homeomorphisms and simplifies existing proofs showing the non-existence of sections of the Mapping Class Group for surfaces of genus greater than 1.

Contribution

It presents a new periodicity criterion and uses it to streamline proofs of non-existence results for sections of the Mapping Class Group.

Findings

The periodicity criterion applies to connected surface homeomorphisms.

Simplified proofs of non-existence of sections for genus > 1 surfaces.

Clarified conditions under which homeomorphisms are periodic.

Abstract

Some years ago, V. Markovic proved that there is no section of the Mapping Class Group for a closed surface of genus g larger than 5 (in the case of homeomorphims) and more recently generalized this result with D. Saric to the case where g is larger than 1. We will state a periodicity criterion and will use it to simplify some of the arguments given by Markovic and Saric in the proof of their theorem. The periodicity criterion tells us that a homeomorphism of a connected surface must be periodic if the set of connected periodic open sets generates the topology of the surface.