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Equivariant topological complexities

Mark Grant

Abstract

Many mechanical systems have configuration spaces that admit symmetries. Mathematically, such symmetries are modelled by the action of a group on a topological space. Several variations of topological complexity have emerged that take symmetry into account in various ways, either by asking that the motion planners themselves admit compatible symmetries, or by exploiting the symmetry to motion plan between functionally equivalent configurations. We will survey the main definitions due to Colman-Grant, Lubawski-Marzantowicz, Błaszczyk-Kaluba and Dranishnikov, and some related notions. We conclude with a short list of open problems.

Equivariant topological complexities

Abstract

Many mechanical systems have configuration spaces that admit symmetries. Mathematically, such symmetries are modelled by the action of a group on a topological space. Several variations of topological complexity have emerged that take symmetry into account in various ways, either by asking that the motion planners themselves admit compatible symmetries, or by exploiting the symmetry to motion plan between functionally equivalent configurations. We will survey the main definitions due to Colman-Grant, Lubawski-Marzantowicz, Błaszczyk-Kaluba and Dranishnikov, and some related notions. We conclude with a short list of open problems.
Paper Structure (7 sections, 36 theorems, 53 equations, 4 figures)

This paper contains 7 sections, 36 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Group actions
  3. Equivariant topological complexity
  4. Invariant topological complexity
  5. Strongly equivariant topological complexity
  6. Effective topological complexity
  7. Problems

Key Result

Proposition 3.3

The equivariant topological complexity is $G$-homotopy invariant. That is, if $X\simeq_G Y$, then ${\sf TC}_G(X)={\sf TC}_G(Y)$.

Figures (4)

  • Figure 1: A configuration in $Y$
  • Figure 2: A configuration in $S^1\times Y$
  • Figure 3: A configuration $x=(\theta_1,\ldots, \theta_6)$ in $X$
  • Figure 4: The symmetric configuration $\tau x$

Theorems & Definitions (81)

  • Example 1.1
  • Example 1.2: from LubMar
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 71 more