Pietro Capovilla, Clara Loeh, Marco Moraschini
Amenable category is a variant of the Lusternik-Schnirelman category, based on covers by amenable open subsets. We study the monotonicity problem for degree-one maps and amenable category and the relation between amenable category and topological complexity.
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Let $A_L$ be the right-angled Artin group associated to a finite flag complex $L$. We show that the amenable category of $A_L$ equals the virtual cohomological dimension of the right-angled Coxeter group $W_L$. In particular, right-angled Artin groups satisfy a question of Capovilla--Löh--Moraschini proposing an inequality between the amenable category and Farber's topological complexity.
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 manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$. For topological complexity we prove the inequality $\TC (M\# N)\ge\max\{\TC M,\TC N\}$ for simply connected manifolds.
On distributional one-category, diagonal distributional complexity, and related invariants
本文提出分布性单类和对角分布性复杂度等新不变量，用于研究非简形空间和空间积上的拓扑性质，并提供了首个超出实射影空间的闭流形例子。
Analog category and complexity
本研究探讨Lusternik–Schnirelmann范畴与拓扑复杂度的概率变体，通过计算揭示其与经典概念的异同，并在非球面情形下推导相关定理对应结论及有限情形的通用上界。
Distributional Topological Complexity and LS-category
本文提出更适合自主系统运动规划的分布拓扑复杂度 dTC 及其对应 dcat，证明其同伦不变性，并对若干空间计算、估计且与 TC、cat 比较。
We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtained an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.
If a map $f$ has a homotopy retraction, then Doeraene and El Haouari conjectured that the sectional category and the relative category of $f$ are the same. In this work we discuss this conjecture for some lower bounds of these invariants. In particular, when we consider the diagonal map, we obtain results supporting Iwase-Sakai's conjecture which asserts that the topological complexity is the monoidal topological complexity.
Up to one approximations of sectional category and topological complexity
本文在统一框架下研究了James的截面类别和Farber的拓扑复杂性，引入了映射的相对和强相对类别概念，证明两者与截面类别相差不超过1，并利用这些工具确定了Ganea纤维的截面和相对类别。
LS-category and topological complexity of Milnor manifolds and corresponding generalized projective product spaces
本文研究了Milnor流形及相应广义射影积空间的LS-范畴与拓扑复杂度，确定了其精确值或紧致界限，并计算了等变情形下的界限及利用mod-2上同调环推导了相关空间的界限。
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.
The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik-Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times\mathbb{R}^n$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.
This survey is a guide for the non specialist on how to use rational homotopy theory techniques to get approximations of Farber's topological complexity, in particular, and of Schwarz's sectional category, in general.
We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.
We compute the Lusternik-Schnirelmann category and the topological complexity of no $k$-equal manifolds $M^{(k)}_d(n)$ for certain values of $d$, $k$ and $n$. This includes instances where $M^{(k)}_d(n)$ is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring $H^*(M^{(k)}_d(n))$ as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.
Axiomatic Sectional category, Topological complexity, and Homotopic distance
本文通过三个公理构建了截面类别、拓扑复杂度和同调距离的公理化框架，并引入了统一这些概念的提升类别概念。
LS-category and topological complexity of several families of fibre bundles
本文针对多类纤维丛及广义射影积空间，利用 Z₂ 上同调环与等变方法，给出其 LS-范畴和拓扑复杂度的紧上界或精确值。
Let $X$ be a two-cell complex with attaching map $α\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1<q\leq3(p-1)$ under fairly mild conditions by making use of the Hopf invariant techniques recently developed by the authors in their study of the sectional category of arbitrar…
Topological complexity for closed 1-forms
引入闭1-形式的拓扑复杂度概念，建立其基本性质与不等式，并证明梯度流方法给出上界。
We give bounds for the module sectional category of products of maps which generalise a theorem of Jessup for Lusternik-Schnirelmann category. We deduce also a proof of a Ganea type conjecture for topological complexity. This is a first step towards proving the Ganea conjecture for topological complexity in the rational context.
We give simple upper bounds for rational sectional category and use them to compute invariants of the type of Farber's topological complexity of rational spaces. In particular we show that the sectional category of formal morphisms reaches its cohomological lower bound and give a method to compute higher topological complexity of formal spaces in terms of their cohomology.
In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously known `topological complexity' of a map. Moreover, we study the relation between Lusternik-Schnirelmann category and topological complexity of two closed oriented manifolds connected by a degree one map.
On properties of effective topological complexity and effective Lusternik-Schnirelmann category
本文研究有效拓扑复杂度的性质，引入有效LS-畴数概念，探究其与群作用轨道映射的关系并给出示例，还提出基于饱和对角线上同调维数界的非消失准则。
Topological Complexity of $S^3/Q_8$ as fibrewise L-S category
本文基于纤维化 L-S 范畴与拓扑复杂度的等价关系，给出确定拓扑球面空间形式拓扑复杂度的算法，并用 Python 代码计算 $S^3/Q_8$ 的拓扑复杂度。
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 more generally for the rational sectional category of a map, in terms of the rational category of a certain auxiliary space. We use our results to deduce consequences for the global (rational) homotopy structure of simply connected, hyperbolic finite complexes.
Combination of open covers with $π_1$-constraints
本文利用可缩 G-CW-复形的稳定化子证明了广义 Lusternik–Schnirelmann 范畴的组合定理，并由此得到流形粘合与循环分支覆盖的单纯体积消失结果以及 Farber 拓扑复杂度的上界。
Relative sectional category revisited
本文探讨了相对截面范畴的扩展及其性质，揭示了其如何统一近期文献中多种同伦数值不变量，如映射拓扑复杂度、相对拓扑复杂度和连续映射的同伦距离。
Sectional category with respect to group actions and sequential topological complexity of fibre bundles
本文引入关于群作用的截面范畴，建立多种 $G$-同伦不变量，并据此给出结构群为 $G$ 的纤维丛序列拓扑复杂度的加性上界及应用。
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 Farber) which has interesting applications in Robotics, specifically, in the robot motion planning problem. In this paper we calculate the Lusternik-Schnirelmann category and as a consequence we calculate the topological complexity of the two-point ordered configuration space of $\mathbb{CP}^n$ for every $n\geq 1$.
We are trying to look over the Lusternik-Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC.
LS-category and sequential topological complexity of symmetric products
本文研究闭曲面及有限CW复形对称积的拓扑性质，计算了相关空间的序列拓扑复杂度与LS-范畴，验证了Farber-Oprea有理性猜想，给出了对应指标的新下界并推导了相关同伦群结论。
Approaches to critical point theory via sequential and parametrized topological complexity
本文通过序列和参数化拓扑复杂性方法，研究临界点理论并建立相关下界，同时计算了特定流形的参数化拓扑复杂性。
We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik-Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the non-equivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the non-equivariant topological complexity.
Proper topological complexity
本文提出Proper拓扑复杂度，要求运动算法避开所有不安全区域，并将其刻画为外部截面范畴的实例，从而成为同伦函子性不变量。
LS-category and topological complexity of real torus manifolds and Dold manifolds of real torus type
本文计算了实 torus 流形和 Dold 流形的 LS 分类，并给出了其拓扑复杂度的精确界，研究了小覆盖的 $\mathbb{Z}_2$-equivariant LS 分类，并确定了对称拓扑复杂度的精确值。
Motion planning and topological complexity for diagrams of spaces
本文针对空间图系统提出拓扑复杂度概念，定义为端点评估映射的截面范畴，通过轨道形状选择实现范畴压缩，并给出上下界，推广了等变情形。
We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied in the study of Farber's topological complexity for 2-cell complexes, as well as in the construction of a counter-example to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category.
Generalized topological complexity and its monoidal version
本文综述了将Srinivasan关于Lusternik-Schnirelmann范畴的任意子集实现结果推广到广义拓扑复杂度及其幺半群版本，并探讨其重要推论。
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 $\TCM{B}$ of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities $\TC{B} = \catBb{\double{B}}+1$ and $\TCM{B} = \catBB{\double{B}}+1$, where $\double{B}=B{\times}B$ is a fibrewise pointed space over $B$ whose proje…
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 system starts at the home configuration and possibly comes back after passing through a list of locations, we define three other classes of topological complexities $\mathbf{LTC}_n(X)$, $\mathbf{ltc}_n(X)$ and $\mathbf{tc}_n(X)$. We will compare these notions and compute the latter for some familiar classes of spaces.
On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces
本文研究了非定向曲面的对称产品的k-th Z_2零除子上同调长度，从而估计并确定了序列拓扑复杂度和LS-范畴，结果恢复了偶维实射影空间和闭非定向曲面的已知事实，并展示了随着g增长，TC_2的行为与其他不变量不同。
Equivariant category and Topological complexity of wedges
本文研究了楔的等变范畴和拓扑复杂度，证明了楔的等变范畴公式，并讨论了相关拓扑复杂度的计算方法。
Intertwining category and complexity
本文提出交织分布版本的LS范畴和序列拓扑复杂性，证明其性质与经典和分布版本相似，并通过Higman群示例展示其不同。
Various Topological Complexities of Small Covers and Real Bott Manifolds
本文研究小覆盖、实矩角流形与实 Bott 流形，计算其 LS-范畴、等变 LS-范畴、拓扑复杂度紧下界、对称拓扑复杂度及 LS 一范畴。
In this paper we analyze some relationships between the topological complexity of a space $X$ and the category of $C_{Δ_X},$ the homotopy cofibre of the diagonal map $Δ_X:X\rightarrow X\times X.$ We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.
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…
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