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As a first goal, it is explained why Goodwillie-Weiss calculus of embeddingsoffers new information about the Euclidean embedding dimension of P^m only form < 16. Concrete scenarios are described in these low-dimensional cases,pinpointing where to look for potential, but critical, high-order obstructionsin the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 istranslated into the triviality of a certain cohomotopy Euler class which, inturn, becomes the only Taylor obstruction to producing an n-dimensionalEuclidean embedding of P^m. A speculative bordism-type form of this primaryobstruction is proposed as an analogue of Davis' BP-approach to the immersionproblem of P^m. A form of the Euler class viewpoint is applied to showTC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spacesfor which the method could produce new information. As a second goal, the paperextends Farber's work on the motion planning problem in order to develop thenotion of a symmetric motion planner for a mechanical system S. FollowingFarber's lead, this concept is connected to the symmetric topologicalcomplexity of the state space of S. The paper ends by sketching theconstruction of a concrete 5-local-rules symmetric motion planner for P^3.

Symmetric topological complexity as the first obstruction in Goodwillie's Euclidean embedding tower for real projective spaces