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

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

2009-11-05

arxiv.org/abs/0911.1116v4

Mathematics

Algebraic Topology

Abstract updated, final version to appear in Transactions of the American Mathematical Society

Scientific paper

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. 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 > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. 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 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.

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