Note on the classification of the orientation reversing homeomorphisms of finite order of surfaces

TL;DR

This paper provides a topological classification of orientation reversing homeomorphisms of finite order on closed surfaces, specifically addressing cases with periods multiple of 4, correcting previous errors in the literature.

Contribution

It offers a corrected and extended classification for orientation reversing homeomorphisms of surfaces with periods multiple of 4, building on prior work and fixing earlier inaccuracies.

Findings

Classification for periods multiple of 4 obtained

Errors in previous classifications corrected

Methodology based on earlier approaches extended

Abstract

The aim of this note is to stablish the topological classification of finite period orientation reversing autohomeomorphims of a closed oriented surface when the period is 2q, with q even. The classification of periodic orientation reversing autohomeomorphims of a closed oriented surface has been made by Kazuo Yokoyama in Complete classification of periodic maps on compact surfaces, Tokyo J. Math. 15 (1992), no. 2, 247--279, and by the author in Classification of the orientation reversing homeomorphisms of finite order of surfaces, Topology and its applications 62 (1995) 145--162 (reference [1] of this paper) following different approaches. In [1] there are some errors for the case considered in this note: orientation reversing homeomorphims of period multiple of 4. The errors have been pointed out by Weibiao Wang of the School of Mathematical Sciences of Peking University and Chao Wang of the School of Mathematical Sciences of East China Normal University in Shanghai. The approach to this problem in [1] is useful and in Section 3 we obtain the classification for the case when the orientation reversing homemorphism has period multiple of 4 following the ideas of [1]. The results in [1] has been used in references [2] and [3]. The last Section include the corrections to these articles following Section 3.