Equivariant embeddings of manifolds into Euclidean spaces (Q2676976)

Given an action of a finite group \(G\) on a compact, connected, smooth manifold \(M\) (or a polyhedron, or a finite graph), it is well-known that there exists a \(G\)-equivariant embedding of \(M\) into Euclidean space \(\mathbb{R}^m\), for some \(m\) (i.e., the \(G\)-action on \(M\) is realized by the restriction of a group \(G \subset SO(m)\) of orthogonal transformations). Then, given a pair \((M,G)\), one may ask for the minimal dimension of such an equivariant embedding into \(\mathbb{R}^m\); it is the main point of the present paper to give some general upper and lower bounds for this minimal dimension. ``First, we provide an upper bound: there exists a \(G\)-equivariant embedding of \(M\) into \(\mathbb{R}^{d|G|+1}\), where \(|G|\) is the order of \(G\) and \(M\) embeds into \(\mathbb{R}^d\). Next we provide a lower bound for a 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 \(\mathbb{R}^m\), then \(m \ge 2l\).'' Some applications to \(G\)-equivariant embeddings of surfaces are given; for example, for any given integer \(m > 0\) there is a periodic map of a closed orientable surface which has no equivariant embedding into \(\mathbb{R}^m\). The case of finite graphs with large automorphism groups is considered in a recent paper by the present author [Discrete Comput. Geom. 67, No. 4, 1257--1265 (2022; Zbl 1490.57030)] where the minimal dimension of an equivariant embedding is determined.