Equivariant embeddings of manifolds into Euclidean spaces

TL;DR

This paper investigates explicit bounds for embedding manifolds with finite group actions into Euclidean spaces, providing both upper and lower bounds, with applications to surfaces.

Contribution

It offers explicit bounds on the dimension of Euclidean spaces needed for equivariant embeddings of manifolds with finite group actions, extending previous existence results.

Findings

Existence of a G-equivariant embedding into R^{d|G|+1}

Lower bounds for cyclic group actions based on orbit lengths

Applications to embedding surfaces with group actions

Abstract

Suppose a finite group $G$ acts on a manifold $M$. By a theorem of Mostow, also Palais, there is a $G$-equivariant embedding of $M$ into the $m$-dimensional Euclidean space $\RR^{m}$ for some $m$. We are interested in some explicit bounds of such $m$. First we provide an upper bound: there exists a $G$-equivariant embedding of $M$ into $\RR^{d|G|+1}$, where $|G|$ is the order of $G$ and $M$ embeds into $\RR^d$. Next we provide a lower bound for finite cyclic group action $G$: If there are $l$ points having pairwise co-prime lengths of $G$-orbits greater than $1$ and there is a $G$-equivariant embedding of $M$ into $\RR^{m}$, then $m\ge 2l$. Some applications to surfaces are given.