On higher analogs of topological complexity

Bulletin of the London Mathematical Society, 2014

We present a new approach to an equivariant version of Farber's topological complexity called invariant topological complexity. It seems that the presented approach is more adequate for the analysis of impact of a symmetry on a motion planning algorithm than the one introduced and studied by Colman and Grant. We show many bounds for the invariant topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the topological complexity of the orbit space. We define the Whitehead version of it.

Algebraic & Geometric Topology, 2014

TURKISH JOURNAL OF MATHEMATICS

The intersection of topological robotics and digital topology leads to us a new workspace. In this paper we introduce the new digital homotopy invariant digital topological complexity number T C(X, κ) for digital images and

Journal of Applied and Computational Topology

It has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence.

2021

In this paper, we examine the relations of two closely related concepts, the digital Lusternik-Schnirelmann category and the digital higher topological complexity, with each other in digital images. For some certain digital images, we introduce κ−topological groups in the digital topological manner for having stronger ideas about the digital higher topological complexity. Our aim is to improve the understanding of the digital higher topological complexity. We present examples and counterexamples for κ−topological groups.

arXiv (Cornell University), 2022

The higher topological complexity of a space X, TC r (X), r = 2, 3,. . ., and the topological complexity of a map f , TC(f), have been introduced by Rudyak and Pavešić, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map f , TC r,s (f), for 1 ≤ s ≤ r ≥ 2, which simultaneously extends Rudyak's and Pavešić's notions. Our unified concept is relevant in the r-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in s prescribed stages of the motion task. We study the homotopy invariance and the behavior of TC r,s under products and compositions of maps, as well as the dependence of TC r,s on r and s. We draw general estimates for TC r,s (f : X → Y) in terms of categorical invariants associated to X, Y and f. In particular, we describe within one the value of TC r,s in the case of the non-trivial double covering over real projective spaces, as well as for their complex counterparts.

Applied General Topology

Y. Rudyak develops the concept of the topological complexity TC(X) defined by M. Farber. We study this notion in digital images by using the fundamental properties of the digital homotopy. These properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. Moreover, we show that the cohomological lower bounds for the digital topological complexity TC(X,κ) do not hold.

Proceedings of the American Mathematical Society, 2018

We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex and replacing the concept of homotopy by that of contiguous simplicial maps. We study the links of this new invariant with those of simplicial category and topological complexity.

arXiv (Cornell University), 2022

We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduced the generalized version of the relative topological complexity of a topological pair on both the Schwarz genus and the homotopic distance. With these concepts, we give some inequalities including the topological complexity and the Lusternik-Schnirelmann category, the most important parts of the study of robot motion planning in topology. Finally, by defining the parametrised topological complexity via the homotopic distance, we present some estimates on the higher setting of this concept.

Cornell University - arXiv, 2020

Digital topology has its own working conditions and sometimes differs from the normal topology. In the area of topological robotics, we have important counterexamples in this study to emphasize this red line between a digital image and a topological space. We indicate that the results on topological complexities of certain path-connected topological spaces show alterations in digital images. We also give a result about the digital topological complexity number using the genus of a digital surface in discrete geometry.