Embedding periodic maps of surfaces into those of spheres with minimal dimensions

TL;DR

This paper determines the minimal dimension into which periodic maps of surfaces can be embedded, revealing that for each integer greater than one, infinitely many maps achieve a specific minimal embedding dimension.

Contribution

It precisely identifies the smallest embedding dimension for periodic maps of surfaces when the order exceeds three times the genus, and shows the existence of infinitely many maps with a given minimal dimension.

Findings

Minimal embedding dimension is determined for certain periodic maps.

Existence of infinitely many maps with a specific minimal dimension.

The minimal dimension can be explicitly characterized for maps with order n ≥ 3g.

Abstract

It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\geq 3g$. We show that for each integer $k>1$ there exist infinitely many periodic maps such that the smallest possible $m$ is equal to $k$.