Related papers: An upper bound for higher topological complexity a…
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 define a new version of Topological Complexity (TC) of a space, denoted as $\text{dTC}$, which, we think, fits better for motion planning for some autonomous systems. Like Topological complexity, \text{dTC} is also a homotopy invariant.…
Given a space $X$, the topological complexity of $X$, denoted by $TC(X)$, can be viewed as the minimum number of "continuous rules" needed to describe how to move between any two points in $X$. Given subspaces $Y_1$ and $Y_2$ of $X$, there…
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…
We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces X whose higher topological complexity TC_n(X) is…
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…
We prove the formula \begin{equation*} \text{cat}_G(X\vee Y)=\max\{\text{cat}_G(X),\text{cat}_G(Y)\} \end{equation*} for the equivariant category of the wedge $X\vee Y$. As a direct application, we have that the wedge $\bigvee_{i=1}^m X_i$…
This paper explores topological complexity in the finite equivariant setting. We first define and study an equivariant version of Tanaka's combinatorial complexity for finite topological spaces. We explore the relationships between this…
We introduce notions of {\it upper chernrank} and {\it even cup length} of a finite connected CW-complex and prove that {\it upper chernrank} is a homotopy invariant. It turns out that determination of {\it upper chernrank} of a space $X$…
The topological complexity ${\sf TC}(X)$ is a homotopy invariant of a topological space $X$, motivated by robotics, and providing a measure of the navigational complexity of $X$. The topological complexity of a connected sum of real…
We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological…
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of…
Digital topological methods are often used on computing the topological complexity of digital images. We give new results on the relation between reducibility and digital contractibility in order to determine the topological complexity of a…
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,…
We use Brown-Peterson cohomology to obtain lower bounds for the higher topological complexity, TC_k(RP^n), of real projective spaces, which are often much stronger than those implied by ordinary mod-2 cohomology.
We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.
In this paper, we associate to two given continuous maps $f,g: X\rightarrow Z$, on a path connected space $X$, the relative topological complexity $TC^{(f, g, Z)}(X):=TC_X(X\times _ZX)$ of their fiber space $X\times _ZX$. When $g=f$ we…
In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for…
For any simply-laced type simple Lie algebra $\mathfrak{g}$ and any height function $\xi$ adapted to an orientation $Q$ of the Dynkin diagram of $\mathfrak{g}$, Hernandez-Leclerc introduced a certain category $\mathcal{C}^{\leq \xi}$ of…
We show how locally smooth actions of compact Lie groups on a manifold $X$ can be used to obtain new upper bounds for the topological complexity $\TC(X)$, in the sense of Farber. We also obtain new bounds for the topological complexity of…