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Symmetric Bi-Skew Maps and Symmetrized Motion Planning in Projective Spaces

Part of: Classical Topics in Algebraic Topology Artificial intelligence (68Txx) Differential topology Operations and obstructions

Published online by Cambridge University Press:  06 August 2018

Jesús González
Jesús González*
Affiliation:
Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Av. IPN 2508, Zacatenco, México City 07000, Mexico ([email protected])
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Abstract

This work is motivated by the question of whether there are spaces X for which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space ${\open R}\hbox{P}^m$, it is known that $\hbox{TC}^S ({\open R}\hbox{P}^{m})$ captures, with a few potential exceptional cases, the Euclidean embedding dimension of ${\open R}\hbox{P}^{m}$. We now show that, for all m≥1, $\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$ is characterized as the smallest positive integer n for which there is a symmetric ${\open Z}_{2}$-biequivariant map Sm×Sm→Sn with a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of ${\open R}\hbox{P}^{2^{e}}$ for e≥1. In particular, this leaves the torus S1×S1 as the only closed surface whose symmetric (symmetrized) TCS (TCΣ) invariant is currently unknown.

Keywords

topological complexitysymmetric motion planningaxial maps with further structureequivariant partition of unitysymmetric square of a space

MSC classification

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space 55S15: Symmetric products, cyclic products
Secondary: 57R40: Embeddings 68T40: Robotics
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1087 - 1100
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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