Related papers: Higher Topological Complexity For Fibrations
We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we apply this construction to a fibration whose fibers are the homotopy categories of the…
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…
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…
This work serves as an opening and basis of an ongoing program investigating topological and geometric aspects of the moduli space of smooth fiberings on a manifold. The present paper focuses on the algebraic and differential topology of…
This paper presents a combinatorial analog of topological complexity for finite spaces. We demonstrate that this coincides with the genuine topological complexity of the original finite space, and constitutes an upper bound for the…
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…
Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected…
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…
We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface…
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 $Y$…
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.
We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc(K) = 4, where K denotes Klein bottle.
We prove the formula \begin{equation*} TC(X\vee Y)=\max\{TC(X),TC(Y),cat(X\times Y)\} \end{equation*} for the topological complexity of the wedge $X\vee Y$.
Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by…
We prove for non-elementary torsion-free hyperbolic groups $\Gamma$ and all $r\ge 2$ that the higher topological complexity ${\sf{TC}}_r(\Gamma)$ is equal to $r\cdot \mathrm{cd}(\Gamma)$. In particular, hyperbolic groups satisfy the…
In this paper, we prove that certain spherical fibrations over certain CW-complexes are stably fibre homotopy equivalent to $\mm{TOP}$-spherical fibrations (see Definition 1,1). Applying this result, we get a sufficient condition for…
The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fibrewise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also…
We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization…
We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover…
In this paper, we study h-fibrations, a weak homotopical version of fibrations which have weak covering homotopy property. We present some homotopical analogue of the notions related to fibrations and characterize h-fibrations using them.…