Equivariant embeddings of manifolds into Euclidean spaces

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 Rm 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 Rd|G|+1, where |G| is the order of G and M embeds into Rd. 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 Rm, then m≥2l.Some applications to surfaces are given.