Discrete Topological Complexities of Simplical Maps

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.

Journal of Symbolic Computation, 2005

This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain explicit combinatorial descriptions of Steenrod kth powers exclusively in terms of face operators.

arXiv: Algebraic Topology, 2019

We undertake a systematic study of the notion of fibration in the setting of abstract simplicial complexes, where the concept of `homotopy' has been replaced by that of `contiguity'. Then a fibration will be a simplicial map satisfying the `contiguity lifting property'. This definition turns out to be equivalent to a known notion introduced by G. Minian, established in terms of a cylinder construction $K \times I_m$. This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration have the same strong homotopy type, a notion that has been recently introduced by Barmak and Minian; any fibration with a strongly collapsible base is fibrewise trivial; and some other ones. We introduce the concept of `simplicial finite-fibration', that is, a map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration $PK \...

2021

We introduce the higher topological complexity (TCn) of a fibration in two ways: the higher homotopic distance and the Schwarz genus. Then we have some results on this notion related to TC, TCn or cat of a topological space or a fibration. We also show that TCn of a fibration is a fiber homotopy equivalence.

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for simplicial homology doesn't be hold in digital images. Moreover we show that the Hurewicz Theorem need not be hold in digital images. Copyright © AJCTA, all rights reserved.

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.

2012

Yu. Rudyak has recently extended Farber’s notion of topological complexity by defining, for n ≥ 2, the nth topological complexity TCn(X) of a path-connected space X—Farber’s original notion is recovered for n = 2. In this paper we develop further the properties of this extended concept, relating it to the Lusternik-Schnirelmann category of cartesian powers of X, as well as to the cup-length of the diagonal embedding X ↪ → Xn. We compute the numerical values of TCn for products of spheres, closed 1-connected symplectic manifolds (e.g. complex projective spaces), and quaternionic projective spaces. We explore the symmetrized version of the concept (TCSn(X)) and introduce a new sym-metrization (TCΣn (X)) which is a homotopy invariant of X. We obtain a (conjecturally sharp) upper bound for TCSn(X) when X is a sphere. This is attained by introducing and studying the idea of cellular stratified spaces, a new concept that allows us to import techniques from the theory of hyperplane arrange...

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

arXiv (Cornell University), 2021

In this study, we improve the topological complexity computations on digital images with introducing the digital topological complexity computations of a surjective and digitally continuous map between digital images. We also reveal differences in topological complexity computations of maps between digital images and topological spaces. Moreover, we emphasize the importance of the adjacency relation on the domain and the range of a digital map in these computations.

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.