Related papers: Topological complexity of configuration spaces
The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound on the number of critical points of a smooth function on a manifold. Another similar invariant is the topological complexity $TC(X)$ (a la…
We compute the higher topological complexity of ordered configuration spaces of orientable surfaces, thus extending Cohen-Farber's description of the ordinary topological complexity of those spaces.
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…
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…
Farber and Rudyak introduced topological complexity $\mathbf{TC}(X)$ of motion planning and its higher analogs $\mathbf{TC}_n(X)$ to measure the complexity of assigning paths to point tuples. Motivated by motion planning where a robotic…
We introduce the geodesic complexity of a metric space, inspired by the topological complexity of a topological space. Both of them are numerical invariants, but, while the TC only depends on the homotopy type, the GC is an invariant under…
We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\rightarrow X_2\rightarrow X_3\rightarrow...$ is bounded above by $2max{TC(X_i); i=1,2,...}$.
We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f\colon X\to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$…
In this paper, we transfer the problem of measuring navigational complexity in topological spaces to the nearness theory. We investigate the most important component of this problem, the topological complexity number (denoted by TC), with…
In this paper, we investigate discrete topological complexity $TC(K)$ introduced for situations where the configuration space possesses a simplicial structure. %Simplicial complexes are well-known and commonly used in programming for…
In this paper, we introduce the notion of transversal topological complexity (TTC) for a smooth manifold $X$ with respect to a submanifold of codimension 1 together with basic results about this numerical invariant. In addition, we present…
We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules…
In this paper we introduce and study a new concept of parametrised topological complexity, a topological invariant motivated by the motion planning problem of robotics. In the parametrised setting, a motion planning algorithm has high…
We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(\pi_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its…
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 study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either…
We define and develop a homotopy invariant notion for the topological complexity of a map $f:X \to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore,…
The Topological complexity a la Farber $\text{TC}(-)$ is a homotopy invariant which have interesting applications in Robotics, specifically, in the robot motion planning problem. In this work we calculate the topological complexity of the…
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 determine topological complexity of a series of finite spaces which is weakly homotopy equivalent to a circle $S^1$, and give a finite space $X$ satisfying the inequality tc$(X) <$ cat$(X {\times} X)$. This answers two conjectures on…