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On properties of effective topological complexity and effective Lusternik–Schnirelmann category

Part of: Artificial intelligence (68Txx) Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  22 January 2025

Zbigniew Błaszczyk ,
Arturo Espinosa Baro  and
Antonio Viruel
Zbigniew Błaszczyk
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland ([email protected])
Arturo Espinosa Baro*
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland ([email protected], [email protected]) (Corresponding author)
Antonio Viruel
Affiliation:
Departamento de Álgebra, Geometria y Topologia, Universidad de Málaga, Campus de Teatinos, s/n, 29071 Málaga, Spain ([email protected])
*
*Corresponding author.
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Abstract

The notion of effective topological complexity, introduced by Błaszczyk and Kaluba, deals with using group actions in the configuration space in order to reduce the complexity of the motion planning algorithm. In this article, we focus on studying several properties of this notion of topological complexity. We introduce a notion of effective LS category which mimics the behaviour the usual LS category has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect to the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.

Keywords

effective LS categoryeffective topological complexityequivariant topological complexitysectional category

MSC classification

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space 68T40: Robotics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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