Symmetric topological complexity of projective and lens spaces

TL;DR

This paper explores the symmetric topological complexity of projective and lens spaces, linking it to embedding problems and extending classical results to broader classes of manifolds.

Contribution

It introduces a symmetrized version of topological complexity that precisely characterizes embedding problems for projective and lens spaces.

Findings

Symmetrized topological complexity captures embedding obstructions.

Extensions to lens spaces and complex projective spaces are discussed.

Clarifies the relationship between axial maps and embeddings.

Abstract

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.