Homology and orientation reversing periodic maps on surfaces

TL;DR

This paper classifies orientation reversing periodic maps on closed surfaces, extending Nielsen's theory for orientation preserving maps, and provides criteria to determine conjugacy classes based on associated data.

Contribution

It introduces a data classification system for orientation reversing periodic maps and establishes criteria for conjugacy, generalizing Nielsen's theory.

Findings

A group of data characterizes each orientation reversing periodic map.

Two maps with the same data are conjugate.

Maps with period ≥ 3g are conjugate to powers of specific maps.

Abstract

In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a group of data for each orientation reversing periodic map such that two periodic maps with the same data must be conjugate to each other. On the other hand, we give the criterion to judge when two different groups of data correspond to the same conjugacy class. As an application of the results of this paper, we shall show that a given orientation reversing periodic map on $\Sigma_g$ with period larger than or equal to $3g$ must be conjugate to the power of a list of particular types of periodic maps.