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, TC($f$) and cat($f$) satisfy the same inequalities as TC($X$) and cat($X$). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC($f$) to studying group homomorphisms $f:H\to G$.
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.
A topological spherical space form is the quotient of a sphere by a free action of a finite group. In general, their homotopy types depend on specific actions of a group. We show that the monoid of homotopy classes of self-maps of a topological spherical space form is determined by the acting group and the dimension of the sphere, not depending on a specific action.
Discrete version of topological complexity of maps
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）；Indian Institute of Technology Madras（马德拉斯理工学院）；MIT Art, Design and Technology University（麻省理工学院艺术设计与技术大学）
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 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 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.
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 $2max{TC(X_i); i=1,2,...}$.
Weighted cup-length calculations in singular cohomology led Farber and Grant in 2008 to general lower bounds for the topological complexity of lens spaces. We replace singular cohomology by K-theory, and weighted cup-length arguments by considerations with biequivariant maps on spheres to improve on Farber-Grant's bounds by arbitrarily large amounts. Our calculations are based on the identification of key elements conjectured to generate the annihilator ideal of the toral bottom class in the ku-homology of the classifying space of a rank-2 abelian 2-group.
It is shown that a surjective monotone map $X\to Y$ between finite $T_0$-spaces induces a surjective map on homology. As such a map turns out to be a sequence of edge contractions in the Hasse diagram of $X$, followed by a homeomorphism, this leads to an explicit relation between the Betti numbers of $X$ to those of $Y$ and the cokernels of the edge contraction maps on the order complexes.
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…
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.
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.
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.
We introduce the topological complexity of the work map associated to a robot system. In broad terms, this measures the complexity of any algorithm controlling, not just the motion of the configuration space of the given system, but the task for which the system has been designed. From a purely topological point of view, this is a homotopy invariant of a map which generalizes the classical topological complexity of a space.
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.
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.
The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m>1$ and n>1; the answer was previously known in the cases m=2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.
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 consider the topological complexity of subgroups of Artin's braid group consisting of braids whose associated permutations lie in some specified subgroup of the symmetric group. We give upper and lower bounds for the topological complexity of such mixed braid groups. In particular we show that the topological complexity of any subgroup of the n-strand braid group which fixes any two strands is 2n-3, extending a result of Farber and Yuzvinsky in the pure braid case. In addition, we generalise our results to the setting of higher topological complexity.
In this paper, we study three relative LS categories of a map and study some of their properties. Then we introduce the `higher topological complexity' and `weak higher topological complexity' of a map. Each of them are homotopy invariants. We discuss some lower and upper bounds of these in invariants and compare them with previously known `topological complexities' of a map.
In this paper we consider the classification of minimal cellular structures of spaces of topological complexity two under some hypotheses on there graded cohomological algebra. This continues the method used by M.Grant et al. in [1].
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.
Polynomial solving algorithms are essential to applied mathematics and the sciences. As such, reduction of their complexity has become an incredibly important field of topological research. We present a topological approach to constructing a lower bound for the complexity of a polynomial-solving algorithm, and give a concrete algorithm to do this in the case that $\mathrm{deg}(f) = 2,3,4$.
We show that the (normalized) topological complexity of the Klein bottle is $4$. We also show that, for any $g\geq 2$, $TC(N_g)=4$. This completes the recent work by Dranishnikov on the topological complexity of non-orientable surfaces.
We introduce fibrewise Whitehead- and fibrewise Ganea definitions of monoidal topological complexity. We then define several lower bounds for the topological complexity, which improve on the standard lower bound in terms of nilpotency of the cohomology ring. The relationships between these lower bounds are studied.
We present upper and lower bounds for symmetrized topological complexity $TC^Σ(X)$ in the sense of Basabe-González-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square $SP^2(X)$. We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Using these results, we calculate the symmetrized topological complexity of all odd spheres.
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new invariant with those of simplicial and topological LS-category.
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…
Farber introduced a notion of topological complexity $\TC(X)$ that is related to robotics. Here we introduce a series of numerical invariants $\TC_n(X), n=1,2, ...$ such that $\TC_2(X)=\TC(X)$ and $\TC_n(X)\le \TC_{n+1}(X)$. For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.
Using known results about their integral cohomology ring, we prove that the topological complexity of the space of n-gons in R^3 with prescribed side lengths equals 2n-5, assuming that the space is nonempty and does not contain any straight-line polygons.
In this paper, we examine how topological complexity, simplicial complexity, discrete topological complexity, and combinatorial complexity compare when applied to models of $S^1$. We prove that the topological complexity of non-minimal finite models of $S^1$ can be less-than-or-equal-to 3, and that the TC of the minimal finite model of any $n$-sphere is equal to 4 for $n \geq 1$. We show the former using properties of the LS-category, and we show the latter by proving that the TC of the non-Hausdorff suspension of any finite connected $T_0$ space is equal to 4. We also prove a result about the…
For a pair of spaces $X$ and $Y$ such that $Y \subseteq X$, we define the relative topological complexity of the pair $(X,Y)$ as a new variant of relative topological complexity. Intuitively, this corresponds to counting the smallest number of motion planning rules needed for a continuous motion planner from $X$ to $Y$. We give basic estimates on the invariant, and we connect it to both Lusternik-Schnirelmann category and topological complexity. In the process, we compute this invariant for several example spaces including wedges of spheres, topological groups, and spatial polygon spaces. In a…
In this paper, we introduce higher symmetric simplicial complexity $SC_n^Σ(K)$ of a simplicial complex $K$ and higher symmetric combinatorial complexity $CC_n^Σ(P)$ of a finite poset $P$. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - González - Rudyak - Tamaki. We prove that the symmetric simplicial complexity $SC_n^Σ(K)$ is equal to symmetric topological complexity $TC_n^Σ(|K|)$ of the geometric realization of $K$ and the symmetric combinatorial complexity $CC_n^Σ(P)$ is equal to symmetric topological complexity $TC_n^Σ(|\mathcal{K}(P)|)$ of the ge…
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.
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.
In this paper we introduce the concepts of higher equivariant and invariant topological complexity; and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.
We study the topological complexity of work maps with respect to some subspaces of the configuration space and a workspace considered as the target set of the motion of robots. The motivation is to optimize and reduce the number of motion planners for work maps. In this regard, we focus on the useful set of works. We check some basic properties of the targeted complexity of maps, such as homotopical invariance, reduction, the product of maps, and so on. Then we compare these targeted complexities, and we find some inequalities in reducing the number of motion planners. We show that the relativ…
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.
