Donald M Davis
We obtain an explicit formula for the best lower bound for the higher topological complexity, TC_k(P^n), of real projective space implied by mod 2 cohomology.
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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.
This article has been withdrawn due to an error in a proof of the main result.
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.
Fibrewise topological complexity of sphere and projective bundles
本文建立了球丛逐纤维拓扑复杂度的稳定同伦理论版本，推广了Farber和Weinberger的结果，并用平行方法对实、复、四元数射影丛证明了类似结论，同时探讨了Farber和Grant的对称化不变量。
We introduce the digital projective product spaces based on Davis' projective product spaces. We determine an upper bound for the digital LS-category of the digital projective product spaces. In addition, we obtain an upper bound for the digital topological complexity of these spaces. We prove the relation between the digital topological complexity of the digital projective product spaces and sum of the digital topological complexity of the digital projective space by associating with the first digital sphere and the digital topological complexity of the remaining digital spheres through an ex…
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。
Symmetric topological complexity of projective and lens spaces
对于实射影空间，（a）欧氏浸入维数，（b）轴映射的存在性，和（c）拓扑复杂性被认为是同一问题的三个方面。但是当涉及到嵌入维数时，Berrick，Feder和Gitler的经典工作在试图将这些流形的欧氏嵌入的存在性与对称轴映射的存在性相鉴别时留下了一个小的不确定性。作为替代方案，我们表明，（c）的对称化版本捕获，在一个尖锐的方式，嵌入问题。讨论了偶挠透镜空间和复射影空间的推广。
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 projective planes, that is, a high genus nonorientable surface, is known to be maximal. We use algebraic tools to show that the analogous result holds for connected sums of higher dimensional real projective spaces.
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 real projective space of dimension $m.$ In particular, we describe a number $r(m)$, which depends on the structure of zeros and ones in the binary expansion of $m$, and with the property that $TC_s(RP^m)$ is given by $sm$ with an error of at most one provided $s \geq r(m)$ and $m \not\equiv 3 \bmod 4$ (the error vanishes for even $m$). The latter fact appear…
Inspired by the analogous result in the algebraic setting (Theorem 1) we show (Theorem 2) that the product $M \times \mathbb{R}P^n$ of a closed and orientable topological manifold $M$ with the $n$-dimensional real projective space cannot be topologically locally flat embedded into $\mathbb{R}P^{m + n + 1}$ for all even $n > m$.
Parametrized topological complexity of bundles of real projective spaces, I
本文研究实射影空间丛的参数化拓扑复杂度，利用Stiefel–Whitney示性类建立新的下界与锐上界，并显式计算多个具体例子。
We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in high contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of non-simply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimens…
This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an application, we characterize the real projective bundles over 2-dimensional small covers by interpreting the fibre sum operation to some combinatorial operation. Finally, we study when the projectivization of Whitney sum of the tautological line bundle and the tangent bundle over real …
In this short note we observe that the higher topological complexity of an iterated connected sum of real projective spaces is maximal possible. Unlike the case of regular TC, the result is accessible through easy mod 2 zero-divisor cup-length considerations.
The p-triviality of stunted projective spaces
本文引入了拓扑流形的$\mathcal P$-平凡性概念，并完整刻画了截断实与复射影空间的$\mathcal P$-平凡性。
As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of …
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 $n\geq 1$.
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.
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.
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.
We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid of the homotopy type of a CW complex and $G'$ a well-pointed grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G'$. This fact has several applications. As the first application, we show that the connecting map $G\rightarrow\mathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $\mathrm{Map}_0(B_nG,BG)\rightarrow\mathrm{Map}(…
High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories
该研究利用Weiss的酉微积分将向量丛问题转化为稳定同伦计算，借助色与等变同伦理论中的广义上同调理论，推导出球面与复射影空间上非平凡稳定平凡向量丛的存在性。
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。
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 $2max{TC(X_i); i=1,2,...}$.
We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduced the generalized version of the relative topological complexity of a topological pair on both the Schwarz genus and the homotopic distance. With these concepts, we give some inequalities including the topological complexity and the Lusternik-Schnirelmann category, the most important parts of the study of robot motion planning in topology. Finally, by defining the parametrised topological complexity via the homotopic distance, we present some estimates on the higher settin…
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 give an upper bound on the topological complexity of varieties $\mathcal{V}$ obtained as complements in $\mathbb{C}^m$ of the zero locus of a polynomial. As an application, we determine the topological complexity of unordered configuration spaces of the plane.
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 invariant and several others already discussed in the literature: Farber's topological complexity, Tanaka's combinatorial complexity, and Colman-Grant's equivariant Lusternik-Schnirelmann category. We find bounds for equivariant combinatorial complexity and for the necessary lengths of equivariant combinatorial motion plannings. We show that the equivariant top…
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 as low as possible, namely n-1.
Here we give a reformulation of a key lemma in the paper [2], "Spaces of Topological Complexity One", which is necessary due to an oversight.
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。
We establish some upper and lower bounds of the rational topological complexity for certain classes of elliptic spaces. Our techniques permit us in particular to show that the rational topological complexity coincides with the dimension of the rational homotopy for some special families of coformal elliptic spaces.
Topological complexity sequences of groups
本文定义了群的拓扑复杂度序列，作为其米尔诺构造的拓扑复杂度序列。该序列是对群拓扑复杂度的内在细化，且适用于无限余维数群。证明了无限余维数群的拓扑复杂度序列弱递增且无界，并确定了有限偶阶群的渐进行为。
Higher Analogues of Discrete Topological Complexity
本研究引入第n阶离散拓扑复杂度，探究其与单纯Lusternik-Schnirelmann畴数的关系及不同维度间的关联，得出其由对应复形几何实现的第n阶通常拓扑复杂度给出的下界，并给出严格不等的实例。
In this paper, we want to discuss the topology of the non-singular hypersurface $Y^{n}$ with complex dimension $n$ in a projective toric manifold $X^{n+1}$. When $n$ is odd, our main results are a decomposition of $Y^{n}\cong Y'\sharp \ s(S^n \times S^n) $ as a connected sum of $s$ copies of $S^n \times S^n$ with a differential manifold $Y'$ such that $b_n (Y')=0$ or 2. When $n$ is even and the degree of $Y$ in $X$ is big enough, we find that $Y$ also admits such a decomposition $Y'\sharp \ s(S^n \times S^n)$, where $Y'$ satisfy $b_n(Y')-|sign(Y')|=b_n(X)\pm sign(H^n(X))$, where $sign(H^n(X))$…
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。
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.
Topological robotics: motion planning in projective spaces
我们研究拓扑机器人的一个基本问题：旋转的一条线，这是固定的旋转关节在一个基点：一个人想把线从其初始位置到最终位置的空间中的连续运动。最终的目标是构造一个算法，一旦给定初始和最终位置，该算法将执行此任务。任何这样的运动规划算法都会存在不稳定性，这是由拓扑原因造成的。第一作者最近提出了研究机器人运动不稳定性的一般方法。对于任何路连通的拓扑空间X，我们都可以将一个数TC（X）联系起来，称为X的拓扑复杂度。这个数字对于运动规划问题是非常重要的：TC（X）决定了X中所有运动规划算法的不稳定性特征。本文研究了实射影空间的拓扑复杂性。特别地，我们计算所有n<24的TC（RP^n）。我们的主要结果是：（对于不同于1，3，7的n）TC（RP^n）的计算问题等价于寻找最小的k，使得RP^n可以浸入欧几里得空间R^{k-1}。
A topological obstruction to existence of quaternionic Plucker map
本文证明了四元数线性代数的一个命题：不存在从四元数Grassmannian到四元数射影空间的连续映射，使得射影空间上重言丛的第一Pontryagin类的拉回等于Grassmannian上重言丛的第一Pontryagin类.
Real topological Hochschild homology of schemes
该研究证明带对合概形的实拓扑Hochschild同调THR满足基变换与Z/2等变平展拓扑下降性质，并给出射影直线及高维射影空间的相关计算，为相关同调理论研究提供了新结果。
Spaces of Rational Loops on a Real Projective Space
证明了从RP（1）到RP（n）的有理映射空间拓扑逼近实射影空间的闭路空间。作为我们的建设的副产品，我们通过粒子空间获得的Kronecker特性（度）的装饰品的解释。
We prove that real projective space RP^{n-3} is homeomorphic to the space of all isometry classes of n-gons in the plane with one side of length n-2 and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.
We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan-Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\textrm{SL}_2(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreduc…
Let $G$ be a compact and connected Lie group and $PU(\mathcal H)$ be the group of projective unitary operators on a separable Hilbert space $\mathcal H$ endowed with the strong operator topology. We study the space $hom_{st}(G, PU(\mathcal H))$ of continuous homomorphisms from $G$ to $PU(\mathcal H)$ which are stable, namely the homomorphisms whose induced representation contains each irreducible representation an infinitely number of times. We show that the connected components of $hom_{st}(G, PU(\mathcal H))$ are parametrized by the isomorphism classes of $S^1$-central extensions of $G$, and…
In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.
Topological classification of generalized Bott towers
若$B$是环面流形，$E$是$B$上复线丛的Whitney和，则$E$的射影化$P(E)$又是环面流形。从$B$作为一个点开始，重复这个构造，我们得到一个复射影丛序列，我们称之为广义Bott塔。证明了如果塔中的顶流形与复射影空间的乘积有相同的上同调环，则塔中的每一纤维化都是平凡的，使得该顶流形与复射影空间的乘积是同构的.这为我们称之为环面流形的上同调刚性问题“如果环面流形的上同调环是同构的，那么环面流形是同构的吗？“我们提供了两个支持上同调刚性问题的结果。
Higher smooth surgery structure sets of complex projective spaces, part II
本文确定复射影空间高阶光滑手术结构集的挠部分，完成低维下其到拓扑版本的遗忘映射描述，并给出在低维复射影空间映射类群中的应用。
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 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.
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 prove that the topological complexity $\mathrm{TC}(π)$ equals $\mathrm{cd}(π\timesπ)$ for certain toral relatively hyperbolic groups $π$.
We prove the formula $TC(G\ast H)=\max\{TC(G), TC(H), cd(G\times H)\}$ for the topological complexity of the free product of discrete groups with cohomological dimension >2.
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.
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 topological complexity for finite spaces raised by K. Tanaka in 2018.
We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.
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