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

Author: Jesús González
Journal: Trans. Amer. Math. Soc. 363 (2011), 6713-6741
MSC (2010): Primary 57R40, 55M30, 55R80, 70E60
Published electronically: June 3, 2011
MathSciNet review: 2833574
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Abstract: As a first goal, it is explained why the Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of $ \mathrm{P}^m$ only for $ m\leq15$. 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\geq16$, the relation $ \mathrm{TC}^S(\mathrm{P}^m)\geq n$ is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an embedding $ \mathrm{P}^m\subset\mathbb{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 $ \mathrm{P}^m$. A form of the Euler class viewpoint is applied to show that $ \mathrm{TC}^S(\mathrm{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 $ {\mathcal S}$. Following Farber's lead, this concept is connected to $ \mathrm{TC}^S(C({\mathcal S}))$, the symmetric topological complexity of the state space of $ {\mathcal S}$. The paper ends by sketching the construction of a concrete $ 5$-local-rules symmetric motion planner for $ \mathrm{P}^3$.

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Additional Information

Jesús González
Affiliation: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del Instituto Politecnico Nacional, Apartado Postal 14-740 México City, C.P. 07000, México

Keywords: Topological complexity, calculus of embeddings, configuration space.
Received by editor(s): February 10, 2010
Received by editor(s) in revised form: August 11, 2010
Published electronically: June 3, 2011
Additional Notes: The author was partially supported by CONACYT Research Grant 102783.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.