Related papers: Topological complexity of the telescope
The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic…
Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose…
Topological complexity was first introduced in 2003 by Michael Farber as a homotopy invariant for a connected topological space X, denoted by TC(X). Although the invariant is defined in terms of elementary homotopy theory using well-known…
Parametrized topological complexity is a homotopy invariant that represents the degree of instability of motion planning problem that involves external constraints. We consider the parametrized topological complexity in the case of…
We study the higher (or sequential) topological complexity $\mathrm{TC}_s$ of manifolds with abelian fundamental group. We give sufficient conditions for $\mathrm{TC}_s$ to be non-maximal in both the orientable and non-orientable cases. In…
We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and…
The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik-Schnirelmann category of X. Given a mechanical system with…
In the present contribution three means of measuring the geometrical and topological complexity of photons' paths in random media are proposed. This is realized by investigating the behavior of the average crossing number, the mean writhe,…
Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version…
The higher topological complexity of a space $X$, $\text{TC}_r(X)$, $r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$, have been introduced by Rudyak and Pave\v{s}i\'{c}, respectively, as natural extensions of…
We calculate the higher topological complexity TC$_s$ for the complements of reflection arrangements, in other words for the pure Artin type groups of all finite complex reflection groups. In order to do that we introduce a simple…
In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of…
We introduce the higher topological complexity (TC_{n}) 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, TC_{n} or cat of a topological space or a…
The $s$-th higher topological complexity of a space $X$, $TC_s(X)$, can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when $X=RP^m$, the…
We introduce fibrewise Whitehead- and fibrewise Ganea definitions of monoidal topological complexity. We then define several lower bounds for the topological complexity, which improve on the standard lower bound in terms of nilpotency of…
Let $p$ be a branched covering of a Riemann surface to the Riemann sphere $\mathbb{P}^1$, with branching set $B \subset \mathbb{P}^1$. We define the complexity of $p$ as infinity, if $\mathbb{P}^1 \setminus B$ does not admit a hyperbolic…
Let X be a (not-necessarily homotopy-associative) H-space. We show that TC_{n+1}(X) = cat(X^n), for n >= 1, where TC_{n+1}(-) denotes the so-called higher topological complexity introduced by Rudyak, and cat(-) denotes the…
We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call…
We obtain an explicit formula for the best lower bound for the higher topological complexity, TC_k(P^n), of real projective space implied by mod 2 cohomology.
The sphere $S^2$ and the torus $T^2$ are the only closed connected surfaces for which higher topological complexities are known (for each $n\in\{2,3,...\}\subset\mathbb{N}$, $\mathrm{TC}_n(S^2)=n$ and $\mathrm{TC}_n(T^2)=2n-2$). This text…