Invariant topological complexity

Lubawski, Wojciech; Marzantowicz, Wacław

doi:10.1112/blms/bdu090

Cited by 15 publications

(31 citation statements)

References 17 publications

(29 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We recall that TC 2 (X) of Definition 3.1 is Farber's TC(X). The following proposition is proved in [LM15].…”

Section: Maps Of Degree 1 and Topological Complexitymentioning

confidence: 94%

Maps of degree one, relative LS category and higher topological complexities

Rudyak,

Sarkar

2020

Preprint

View full text Add to dashboard Cite

In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce 'higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously known 'topological complexity' of a map. Moreover, we study the relation between Lusternik-Schnirelmann category and topological complexity of two closed oriented manifolds connected by a degree one map.

show abstract

“…We recall that TC 2 (X) of Definition 3.1 is Farber's TC(X). The following proposition is proved in [LM15].…”

Section: Maps Of Degree 1 and Topological Complexitymentioning

confidence: 94%

Maps of degree one, relative LS category and higher topological complexities

Rudyak,

Sarkar

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…equivalence is often described by the action of some group G on C and there are several versions of equivariant topological complexity -see [6], [20], [10] or [3]. Some of them require motion plans to be equivariant maps defined on invariant subsets of C×C, while other consider arbitrary paths that are allowed to 'jump' within the same orbit (see [3, Section 2.2] for an overview and comparison of different approaches).…”

Section: Complexity Of a Mapmentioning

confidence: 99%

A topologist’s view of kinematic maps and manipulation complexity

Pavešić¹

2018

Contemporary Mathematics

View full text Add to dashboard Cite

In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss mechanical devices that consist of rigid parts connected by joints and show how the geometry of the joints determines the forward kinematic map that relates the configuration of joints with the pose of the end-effector of the device. We explain how to compute the dimension of the joint space and describe topological obstructions for a kinematic map to be a fibration or to admit a continuous section. In the second part of the paper we define the complexity of a continuous map and show how the concept can be viewed as a measure of the difficulty to find a robust manipulation plan for a given mechanical device. We also derive some basic estimates for the complexity and relate it to the degree of instability of a manipulation plan.

show abstract

“…Again, the minimal number of atoms is an invariant. Equivariant topological complexity was studied in [7] and, in the modified form of invariant topological complexity, in [13]. Our main result is that, for compact Lie group actions, equivariant LS category (even in the extended sense of [5]) and invariant topological complexity are Morita invariant and therefore suitable as invariants of orbifolds.…”

Section: Introductionmentioning

confidence: 99%

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological Complexity

Ángel¹,

Colman²,

Grant³

et al. 2019

Preprint

View full text Add to dashboard Cite

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant A-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.

show abstract

Invariant topological complexity

Cited by 15 publications

References 17 publications

Maps of degree one, relative LS category and higher topological complexities

Maps of degree one, relative LS category and higher topological complexities

A topologist’s view of kinematic maps and manipulation complexity

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological Complexity

Contact Info

Product

Resources

About