Melih Is, Ismet Karaca
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 fibration. We also show that TC_{n} of a fibration is a fiber homotopy equivalence.
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In this note we define fibrations of topological stacks and establish their main properties. We prove various standard results about fibrations (fiber homotopy exact sequence, Leray-Serre and Eilenberg-Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. When restricted to topological spaces our notion of fibration coincides with the clas…
Parametrized topological complexity of spherical fibrations over spheres
本文研究了球面纤维化的参数化拓扑复杂度，通过弱范畴计算下界并确定了某些纤维化的具体值。
We study Hurewicz fibrations between finite T$_0$--spaces from a combinatorial viewpoint and give strong conditions that a continuous map between finite T$_0$--spaces must satisfy in order to be a Hurewicz fibration. We also show that there exists a strong relationship between Hurewicz fibrations between finite T$_0$--spaces and Grothendieck bifibrations. Finally we give several interesting examples that illustrate this theory and show that many of the assumptions of our results are necessary.
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 finitely generated torsion-free nilpotent groups.
In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.
The fibration sequences of the lattice path operad
本文引入复杂度为$m$的operad概念，证明其对应operad是格路径operad第$m$阶滤过的子operad，并构造相关纤维化序列，推广了已有结果。
Topological complexity of Milnor fibration
本文将 Milnor 纤维化理论与拓扑机器人学结合，研究解析映射芽的 Milnor 纤维化的任务规划问题，发现规划算法依赖纤维化定理，并推测其拓扑复杂度与基一致。
On the topological complexity of directed parametrized motion planning
本文研究了有向参数化运动规划的拓扑复杂度，引入了相关概念并探讨了其与拓扑机器人学中其他不变量的关系。
Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory of principal bundles. Thus, Goerss-Jardine appealed to topological methods for the verification. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use the classifying theory of principal bundles or appeal to topological methods.
Topological Complexity Related To Multi-Valued Functions
本文研究多值函数理论中的机器人运动规划问题，通过引入多同伦提升性质和多纤维化丰富多同伦研究，并计算了空间或多纤维化的拓扑多复杂度与Lusternik-Schnirelmann多维数。
Equivariant and invariant parametrized topological complexity
本文研究了G-等变纤维的不变参数拓扑复杂性，推广了Lubawski和Marzantowicz的不变拓扑复杂性，并计算了等变Fadell-Neuwirth纤维的不变参数拓扑复杂性，用于运动规划问题。
We study formal counterparts of Hurewicz fibrations and related topological notions in elementary toposes with NNO. The constructions are based on a specific notion of interval and lead to a structure of category of fibrant objects on toposes equipped with such a datum. We get in fact slightly more as the building blocks are derived from a weak factorisation system.
Rational sequential parametrized topological complexity
本文从有理同伦理论角度研究序列参数化拓扑复杂度，在纤维丛允许某种分解时给出了显式代数上界，推广了已有拓扑复杂度结果。
Schwarz genus $\mathsf{g}(ξ)$ of a fibration $ξ\colon E\to B$ is defined as the minimal integer $n$, such that there exists a cover of $B$ by $n$ open sets that admit partial section to $ξ$. Many important concepts, including Lusternik-Schnirelmann category, Farber's topological complexity and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain type of morphisms between fibrations. Our main result is the following: if there exist a fibrewise map $f\colon E\to E'$ bet…
Equivariant parametrized topological complexity
本文定义并研究了参数化拓扑复杂度的等变类似物，推广了非等变情形的若干结果，给出了其纤维wise等变同伦不变性及若干界，并计算了特定等变纤维化的该复杂度。
We extend the functor Sing of singular chains to the category of topological stacks and establish its main properties. We prove that Sing respects weak equivalences and takes a morphism of topological stacks that is both a Serre and a Reedy fibration to a Kan fibration of simplicial sets. When restricted to the category of topological spaces Sing coincides with the usual singular functor.
In this paper we develop theory of sequential parametrized motion planning which generalises the approach of parametrized motion planning, which was introduced recently in [3]. A sequential parametrized motion planning algorithm produced a motion of the system which is required to visit a prescribed sequence of states, in certain order, at specified moments of time. The sequential parametrized algorithms are universal as the external conditions are not fixed in advance but rather constitute part of the input of the algorithm. The second part of this article consists of a detailed analysis of t…
Group actions and higher topological complexity of lens spaces
该研究得到纤维化全空间高阶拓扑复杂度的上界，结合高阶子空间拓扑复杂度、紧李群作用改进相关上界，并据此算出多种情形下高维透镜空间高阶拓扑复杂度的精确值。
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 introduce the pointed version of parametrized topological complexity. Finally we give sufficient conditions so that both notions agree.
In their previous work, Barraud and Cornea enriched the Lagrangian Floer complex by adding cubical chains in the based loop space of the Lagrangian, and recovered the Leray-Serre spectral sequence of the based path space fibration, assuming that the Lagrangian is weakly exact and simply connected. In the present article, we remove the simple connectivity assumption and adapt the construction to any Hurewicz fibration. To this end, we introduce a stronger notion of local systems than the classical one, as a topological functor from the free path space of a manifold to the category of topologica…
We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces $X$ for which the topological complexity $\TC(X)$ (defined to be the genus of the free path fibration on $X$) is greater than the zero-divisors cup-length plus one.
By analogy with the invariant Q-category defined by Scheerer, Stanley and Tanré, we introduce the notions of Q-sectional category and Q-topological complexity. We establish several properties of these invariants. We also obtain a formula for the behaviour of the sectional category with respect to a fibration which generalizes the classical formulas for Lusternik-Schnirelmann category and topological complexity.
We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised topological complexity of epimorphisms. Several applications are given, including an alternative computation of the parametrised topological complexity of the planar Fadell--Neuwirth fibrations which avoids calculations involving cup products. We also prove a homotopy invariance result for parametrised topological complexity of fibrations over different bases.
Classification of metric fibrations
本文基于富集范畴视角研究度量纤维化，证明其大小分解性质，将其归为富集Grothendieck纤维化的限制，通过主纤维化、1-Čech上同调、挠子及度量空间基本群完成分类，类比拓扑纤维丛与覆盖空间。
We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to a new and unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic K-theory invariant…
The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's $Γ$-spaces. The formalism of topological operads generalises well to different categories yielding such notions as $\mathbb E_n$-algebras in chain complexes, while the $Γ$-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding "weak" sections of a homotopical Grothendieck fibration. We propose a model for such sections, c…
Dislocations and Fibrations: The Topological Structure of Knotted Smectic Defects
本研究探究近晶液晶中扭结缺陷的拓扑性质，分析螺位错、刃型位错等缺陷与扭结纤维化的关联，给出点缺陷数量下界，深化了与焦锥畴的类比，为相关拓扑理论提供研究平台。
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 Serre path fibration, not many examples are known to be determined concretely by now. In 2010, Iwase and Sakai showed that the topological complexity of a space is a fibrewise version of a L-S category for a fibrewise space over the space. In this paper, we determine the topological complexity of $S^3/Q_8$ using a method produced from the fibrewise view point…
Discrete Topological Complexities of Simplical Maps
该研究聚焦满单纯纤维化的离散拓扑复杂度，分析其与离散几何结构、计算效率的关联及高维性质，还探讨高邻接距离性质，提出可简化离散TC的计算方式。
Approximate Fibrations in Higher Topos Theory
本文在更高拓扑论框架中研究近似纤维化，通过形状理论和拓扑论方法推广并特征化细胞状映射。
Parametrized motion planning algorithms have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly co…
Sequential parametrized topological complexity of group epimorphisms
本文引入群满同态的序列参数化拓扑复杂度，推广了序列拓扑复杂度，利用子群上同调维数给出其界，并应用于平面Fadell-Neuwirth纤维化的计算。
Fibrations and coset spaces for locally compact groups
本文证明当闭子群 $L$ 为局部紧 pro-Lie 群时，陪集空间映射 $G/K\to G/L$ 是纤维化，并由此推导出 Skljarenko、Madison 和 Mostert 的两个经典结论。
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 $Y$ of a robot arm or a similar mechanism. Then one can associate to $f$ a number $\mathrm{TC}(f)$, which is, roughly speaking, the minimal number of continuous rules that are necessary to construct a complete manipulation algorithm for the device. Examples show that $\mathrm{TC}(f)$ is very sensitive to small perturbations of $f$ and that its value depends hea…
A study of the Ganea conjecture for topological complexity by using weak topological complexity
该研究借助 Berstein-Hilton 弱拓扑复杂度及其稳定形式，给出空间满足拓扑复杂度 Ganea 猜想的充分条件，并通过若干例子加以说明。
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.
The need for the extra dimension in Kustaanheimo-Stiefel (KS) regularization is explained by the topology of the Hopf fibration, which defines the geometry and structure of KS space. A trajectory in Cartesian space is represented by a four-dimensional manifold, called the fundamental manifold. Based on geometric and topological aspects classical concepts of stability are translated to KS language. The separation between manifolds of solutions generalizes the concept of Lyapunov stability. The dimension-raising nature of the fibration transforms fixed points, limit cycles, attractive sets, and …
Monoidal and symmetrized parametrized topological complexity
该研究引入并研究了参数化拓扑复杂性的幺半与对称化版本，构建了相关理论的参数化类似物，证明两类对称化参数化拓扑复杂性概念一致，并对Fadell–Neuwirth纤维化计算了其不变量。
We prove an upper bound of higher topological complexity $TC_n(X)$ using higher $\mathcal{D}$-topological complexity $TC_n^{\mathcal{D}}(X)$ of a space $X$. An intermediate invariant $\widetilde{TC}_n(X)$ is used in the proof. We interpret this invariant $\widetilde{TC}_n(X)$ as higher analogue of strongly equivariant topological complexity of the universal cover of $\widetilde{X}$ with the action of the fundamental group of $X$.
We determine the symmetrized topological complexity of the circle, using primarily just general topology.
In their 2012 paper, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer-Hopf conjecture in the complex pro…
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.
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.
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 $2max{TC(X_i); i=1,2,...}$.
Sequential topological complexity of aspherical spaces and sectional categories of subgroup inclusions
本文将非球面空间TC的拓扑机器人学结果推广至诱导子群包含的纤维化截面范畴，给出序列TC与满同态参数化TC的新下界，并推广Costa-Farber典范类以研究更一般空间。
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 topological complexity of its order complex.
We prove Farber's conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity.
We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(π_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.
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