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- W4297492468 abstract "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." @default.
- W4297492468 created "2022-09-29" @default.
- W4297492468 creator A5070099905 @default.
- W4297492468 date "2009-11-05" @default.
- W4297492468 modified "2023-10-18" @default.
- W4297492468 title "Symmetric topological complexity as the first obstruction in Goodwillie's Euclidean embedding tower for real projective spaces" @default.
- W4297492468 doi "https://doi.org/10.48550/arxiv.0911.1116" @default.
- W4297492468 hasPublicationYear "2009" @default.
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