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

Abstract

This paper explains why Goodwillie-Weiss calculus of embeddings can offer new information about the Euclidean embedding dimension of P m only for m � 15. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential—but critical—high-order obstructions in the corresponding Taylor towers. For m � 16, the relation TC S (P m ) � n is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an embedding P m � R n . A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis’ BP-approach to the immersion problem of P m . A form of the Euler class viewpoint is applied to show TC S (P 3 ) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber’s work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber’s lead, this concept is connected to TC S (C(S)), the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P 3 .