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

Symmetric topological complexity of projective and lens spaces