  For real projective spaces, (a) the Euclidean immersion dimension, (b) the
existence of axial maps, and (c) the topological complexity are known to be
three facets of the same problem. But when it comes to embedding dimension, the
classical work of Berrick, Feder and Gitler leaves a small indeterminacy when
trying to identify the existence of Euclidean embeddings of these manifolds
with the existence of symmetric axial maps. As an alternative we show that the
symmetrized version of (c) captures, in a sharp way, the embedding problem.
Extensions to the case of even torsion lens spaces and complex projective
spaces are discussed.
