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

Zbigniew Błaszczyk, Arturo Espinosa Baro, Antonio Viruel

TL;DR

A notion of effective LS category is introduced which mimics the behaviour the usual LS category has in the non-effective setting and is used to investigate the relationship between these effective invariants and the orbit map with respect to the group action.

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 such notion of topological complexity. We introduce a notion of effective LS-category which mimics the behaviour the usual LS-cat has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect of the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.

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

TL;DR

A notion of effective LS category is introduced which mimics the behaviour the usual LS category has in the non-effective setting and is used to investigate the relationship between these effective invariants and the orbit map with respect to the group action.

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 such notion of topological complexity. We introduce a notion of effective LS-category which mimics the behaviour the usual LS-cat has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect of the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.
Paper Structure (8 sections, 28 theorems, 109 equations, 1 figure)

This paper contains 8 sections, 28 theorems, 109 equations, 1 figure.

Table of Contents

  1. Review of effective TC
  2. The global effective path space
  3. Effective LS-category
  4. The problem of ${\rm TC}^{G,\infty}(X) = 0$
  5. Effective topological complexity and the orbit projection
  6. Orbit map has a strict section.
  7. Orbit map is a fibration.
  8. Cohomological conditions for non-vanishing of ${\rm TC}^{G,2}(X)$

Key Result

Theorem 1

Let $X$ be a $G$-space. The following statements hold:

Figures (1)

  • Figure 1: A mechanical arm in physically different, but functionally equivalent states, since grips A and B are indistinguishable.

Theorems & Definitions (61)

  • Theorem : Theorem \ref{['CatTCEffIneq']}, Proposition \ref{['Catmap']} and Corollary \ref{['catTC0']}
  • Theorem : Theorem \ref{['EfOrbSect']} and Theorem \ref{['EffOrbFibr']}
  • Theorem : Theorem \ref{['PropDimDiag1']}, Corollary \ref{['TC2great0']}
  • Theorem 1.1
  • Definition 1.2
  • Lemma 1.3: BlKa2
  • Definition 1.4
  • Theorem 1.5: BlKa2
  • Remark 1.6
  • Lemma 2.1
  • ...and 51 more