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We define and develop a homotopy invariant notion for the sequential topological complexity of a map f : X → Y , denoted T C r (f) , that interacts with T C r (X) and T C r (Y) in the same way Jamie Scott's topological complexity map T C (f) interacts with T C (X) and T C (Y). Furthermore, we apply T C r (f) to studying group homomorphisms ϕ : Γ → Λ. In addition, we give the characterization of cohomological dimension of group homomorphisms. [ABSTRACT FROM AUTHOR]

Let Crit M denote the minimal number of critical points (not necessarily non-degenerate) on a closed smooth manifold M. We are interested in the evaluation of Crit. It is worth noting that we do not know yet whether Crit M is a homotopy invariant of M. This makes the research of Crit a challenging problem. In particular, we pose the following question: given a map f : M → N of degree 1 of closed manifolds, is it true that Crit M ≥ Crit N ? We prove that this holds in dimension 3 or less. Some high dimension examples are considered. Note also that an affirmative answer to the question implies the homotopy invariance of Crit; this simple observation is a good motivation for the research. [ABSTRACT FROM AUTHOR]

We prove the equality cat (ϕ) = cd (ϕ) for epimorphisms ϕ : Γ → Λ between torsion-free, finitely generated almost nilpotent groups Γ and Λ. In addition, we prove the equality cat (ϕ) = cd (ϕ) for homomorphisms ϕ : Γ → Λ between torsion-free, finitely generated virtually nilpotent groups. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract In this article we study the higher topological complexity TC r (X) in the case when X is an aspherical space, X = K (π , 1) and r ≥ 2. We give a characterisation of TC r (K (π , 1)) in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper [7] , joint with M. Grant and G. Lupton, treats the special case r = 2. We also obtain in this paper useful lower bounds for TC r (π) in terms of cohomological dimension of subgroups of π × π × ... × π (r times) with certain properties. As an illustration of the main technique we find the higher topological complexity of Higman's group. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in [16] by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the TC -generating function ∑ r = 1 ∞ TC r + 1 (X) x r encoding the values of the higher topological complexity TC r (X) for all values of r. We show that in many examples (including the case when X = K (H , 1) with H being a RAA group) the TC -generating function is a rational function of the form P (x) (1 − x) 2 where P (x) is an integer polynomial with P (1) = cat (X). [ABSTRACT FROM AUTHOR]

Abstract In [11] , a new approximating invariant TC D for topological complexity was introduced called D -topological complexity. In this paper, we explore more fully the properties of TC D and the connections between TC D and invariants of Lusternik–Schnirelmann type. We also introduce a new TC -type invariant TC ˜ that can be used to give an upper bound for TC , (1) TC (X) ≤ TC D (X) + TC ˜ (X). This then entails a connectivity-dimension estimate TC (X) ≤ TC D (X) + ⌈ 2 dim X − k k + 1 ⌉ , where X is a finite dimensional simplicial complex with k -connected universal cover X ˜. The above inequality is a refinement of an estimate given by Dranishnikov [5]. [ABSTRACT FROM AUTHOR]

We define the m -homotopic distance between maps. We study its properties and we compare it with other known invariants like the m -category and the diagonal topological complexity. [ABSTRACT FROM AUTHOR]

Abstract For a pair of spaces X and Y such that Y ⊆ 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 addition, we connect the invariant to the existence of certain types of axial maps. [ABSTRACT FROM AUTHOR]

This paper presents a notion of Lusternik–Schnirelmann category for small categories, which is an invariant under homotopy equivalences based on natural transformations. We focus on the relationship between this categorical Lusternik–Schnirelmann category and the classical one via the classifying space. We provide a combinatorial method to calculate the classical Lusternik–Schnirelmann category of the classifying space of a finite acyclic category, taking the barycentric subdivision into account. Moreover, we establish the product inequality for fibered and cofibered categories as an analogue of the inequality of the classical Lusternik–Schnirelmann category for Hurewicz fibrations. [ABSTRACT FROM AUTHOR]

We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduce the generalized version of the relative topological complexity of a topological pair with respect to 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. Later, by defining the parametrized topological complexity via the homotopic distance, we present some estimates on the higher setting of this concept. Finally, we give some important examples of the parametrized topological complexities of fiber bundles with their fibers. [ABSTRACT FROM AUTHOR]

The configuration space of n labeled disks of radius r inside the unit disk is denoted Conf n , r ( D 2 ) . We study how the cohomology of this space depends on r . In particular, given a cohomology class of Conf n , 0 ( D 2 ) , for which r does its restriction to Conf n , r ( D 2 ) vanish? A related question: given the configuration space Seg n , r ( D 2 ) of n labeled, oriented segments of length r , it has a map to ( S 1 ) n that records the direction of each segment. For which r does this angle map have a continuous section? The paper consists of a collection of partial results, and it contains many questions and conjectures. [ABSTRACT FROM AUTHOR]

Given a map f : M → N of degree 1 of closed manifolds, is it true that the Lusternik–Schnirelmann category of the range of the map is not more than the category of the domain? We discuss this and some related questions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Lusternik–Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including its relevance to simplicial products and fibrations. We relate this category of a map to the classical Lusternik–Schnirelmann category of a map between finite topological spaces. Finally, we show how the simplicial category of a map may be used to define and study a simplicial version of the category weight of Y. Rudyak and J. Strom. [ABSTRACT FROM AUTHOR]

We study the behavior of the abstract sectional category in the Quillen, the Strøm and the Mixed proper model structures on topological spaces and prove that, under certain reasonable conditions, all of them coincide with the classical notion. As a result, the same conclusions hold for the abstract Lusternik–Schnirelmann category and the abstract topological complexity of a space. [ABSTRACT FROM AUTHOR]

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 invariants and compare them with previously known 'topological complexities' of a map. [ABSTRACT FROM AUTHOR]

In this paper we establish a natural definition of Lusternik–Schnirelmann category for simplicial complexes via the well known notion of contiguity. This simplicial category has the property of being invariant under strong equivalences, and it only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of simplicial geometric category. We prove that the maximum value over the simplicial homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of LS-category are obtained in the setting of finite topological spaces. [ABSTRACT FROM AUTHOR]

We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A × B , whenever A and B are subgroups of π 1 ( X ) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π 1 ( X ) . This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zero-divisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group. [ABSTRACT FROM AUTHOR]

We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate TC ( X ) ≤ 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. [ABSTRACT FROM AUTHOR]

Abstract: We extend the theory of the Lusternik–Schnirelmann category to general metric spaces by means of covers by arbitrary subsets. We also generalize the definition of the strict category weight. We show that if the Bockstein homomorphism on a metric space X is non-zero, then , and use this to compute the category of Pontryagin surfaces. Additionally, we prove that a Polish space with LS-category n can be presented as the inverse limit of ANR spaces of category at most n. [Copyright &y& Elsevier]

Abstract: In this paper we compute: the Schwarz genus of the Stiefel manifold with respect to the action of the Weyl group , and the Lusternik–Schnirelmann category of the quotient space . Furthermore, these results are used in estimating the number of critically outscribed parallelotopes around a strictly convex body, and Birkhoff–James orthogonal bases of a normed finite dimensional vector space. [Copyright &y& Elsevier]

Abstract: We define the LS-category by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik–Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman–Whitehead theorem and Dranishnikovʼs theorem. We compute for some fractal Peano continua such as Menger spaces and Pontryagin surfaces. [Copyright &y& Elsevier]

Abstract: In this paper we give detecting elements for the proper Lusternik–Schnirelmann category of any missing boundary 3-manifold. As a consequence, we determine the proper LS category of all orientable one-ended semistable open 3-manifolds. These results extend those in Cárdenas et al. (2012) [10]. [Copyright &y& Elsevier]

Abstract: There is a problem with the proof of Theorem 1.13 of Iwase and Sakai (2010) which states that for a fibrewise well-pointed space X over B, we have and that for a locally finite simplicial complex B, we have . While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants and is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12. [Copyright &y& Elsevier]

Abstract: We show that , where is Foxʼs m-dimensional category, is the mth Postnikov section of X and is the Lusternik–Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature. [Copyright &y& Elsevier]

Abstract: A closed topological n-manifold is of (resp. )-category 2 if it can be covered by two open subsets , such that the inclusions factor homotopically through maps (resp. ). We characterize all closed n-manifolds of -category 2 and the fundamental groups of all closed n-manifolds of -category 2. [Copyright &y& Elsevier]

Abstract: A 7-dimensional CW-complex having Lusternik–Schnirelmann category equal to 2 is constructed. Using a divisibility phenomenon for Hopf invariants, it is proved that the Cartesian product of the constructed complex with a sphere of sufficiently large dimension also has category 2. This space hence constitutes the minimum dimensional known counterexample to Ganea''s conjecture on the Lusternik–Schnirelmann category of spaces. [Copyright &y& Elsevier]

Abstract: This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category of stratified spaces, that are topological spaces X endowed with a partition and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element of together with a class of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe. If , we define a transverse subset as a subspace A of X such that the intersection is at most countable for any . Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a -foliation, the three previous definitions, with the class of transverse subsets, coincide with the tangential category and are homotopical invariants. [Copyright &y& Elsevier]

Abstract: Farber introduced a notion of topological complexity that is related to robotics. Here we introduce a series of numerical invariants ,  , such that and . For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity. [Copyright &y& Elsevier]

Abstract: Topological complexity 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 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 and , where is a fibrewise pointed space over B whose projection and section are given by the canonical projection to the second factor and the diagonal. In addition, our method in studying fibrewise L–S category is able to treat a fibrewise space with singular fibres. [Copyright &y& Elsevier]

Abstract: We show that the Lusternik–Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G. [Copyright &y& Elsevier]

Abstract: We give a lower bound for the Lusternik–Schnirelmann category of by computing the category weight and the module category weight of . [Copyright &y& Elsevier]

Abstract: In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly p-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel''skii–Schwarz, and the equivariant Lyusternik–Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced. [Copyright &y& Elsevier]

Using the surgery we prove the following: Theorem 0.1 Let f : M → N be a normal map of degree one between closed smooth manifolds with N being (r − 1) -connected, r ⩾ 1. If N satisfies the inequality dim ⁡ N ⩽ 2 r cat (N) − 3 , then cat (M) ⩾ cat (N). [ABSTRACT FROM AUTHOR]

Abstract: We will provide a lower bound for arbitrary proper actions in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant LS-category of its generalized Weyl group. As an application we reprove a theorem of Singhof that determines the classical Lusternik–Schnirelmann category for and . [Copyright &y& Elsevier]

Abstract: We determine the Lusternik–Schnirelmann category of real Stiefel manifolds and quaternionic Stiefel manifolds for which is equal to the cup-length of the mod 2 cohomology of and the integer cohomology of , respectively. [Copyright &y& Elsevier]

Abstract: A general study is undertaken of product-wedge-diagonal (=PWD) structures on a space. In part this concept may be viewed as arising from G.W. Whitehead''s fat-wedge characterization of Lusternik–Schnirelmann category. From another viewpoint PWD-structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD-structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map . A principal result of the paper uses Hopf invariants to formulate a Berstein–Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join that has PWD-structure defined canonically via the join structure in terms of diagonal maps on U and V. [Copyright &y& Elsevier]

Abstract: Motivated by the concept of -category of a manifold introduced by Clapp and Puppe, we give a different proof of a (slightly generalized) Theorem of Hempel and McMillan: If M is a closed 3-manifold that is a union of three open punctured balls then M is a connected sum of and -bundles over . [Copyright &y& Elsevier]

Abstract: The Lusternik–Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used for giving upper-bound for Lusternik–Schnirelmann categories of topological spaces. Singhof has determined the Lusternik–Schnirelmann categories of the unitary groups. In this paper I give two cone-decompositions of each unitary group for alternative proofs of Singhof''s result. One cone-decomposition is easy. The other is closely related to Miller''s filtration and Yokota''s cellular decomposition of the unitary groups. [Copyright &y& Elsevier]

Abstract: This paper presents a study of one-ended locally finite CW-complexes with proper L–S category ⩽2. We detect the class of towers of groups which can be the fundamental pro-group of a space of proper L–S category 2. A second part of the paper is concerned with two-dimensional CW-complexes. For these, we give different characterizations of spaces with proper L–S category ⩽2. [Copyright &y& Elsevier]

Abstract: It is shown that the proper L-S category of an eventually end-irreducible, -irreducible Whitehead 3-manifold is 4. For this we prove, in the category of germs at infinity of proper maps, a partial analogue of the characterization by Eilenberg and Ganea of the L-S category of an aspherical space. [Copyright &y& Elsevier]

Abstract: Let be a fibre bundle with structure group G, where B is -connected and of finite dimension, . We prove that the strong L–S category of X is less than or equal to , if F has a cone decomposition of length m under a compatibility condition with the action of G on F. This gives a consistent prospect to determine the L–S category of non-simply connected Lie groups. For example, we obtain for all , which might be best possible, since we have for any prime p and . Similarly, we obtain the L–S category of for and . We remark that all the above Lie groups satisfy the Ganea conjecture on L–S category. [Copyright &y& Elsevier]

Most of the properties of the category of Lusternik–Schnirelmann come from the cube theorems of Mather, especially the second. The duals of these theorems are false, which makes the dual category problematic. However, a weaker version of the second cube theorem is sufficient to get the usual properties of the LS-category. In this note, we show that this weaker version of the dual of the first theorem is false. The weaker version of the dual of the second cube theorem remains an open problem. [Copyright &y& Elsevier]

The transverse Lusternik–Schnirelmann category of a foliation is an invariant of foliated homotopy type. In this paper we show that the category of a Riemannian foliation is infinite if there exists a non-compact leaf verifying certain conditions. Examples of the obstruction to construct categorical coverings in such foliations are given. [Copyright &y& Elsevier]

We give a cellular decomposition of the compact connected Lie group Spin(7). We also determine the L–S categories of Spin(7) and Spin(8). [Copyright &y& Elsevier]

In this paper we make explicit the relationship between the category of a connected CW complex and the category of its skeleta. In particular we prove that if Y is a connected non-contractible CW complex, then catYk⩽catY for each k-skeleton Yk. On the other hand, we prove that cat0(Y)=limk→∞ cat0(Yk), where cat0 denotes the rational category of the space. [Copyright &y& Elsevier]

We discuss the monotonicity problem for the amenable category of closed manifolds under maps of non-zero degree. In dimension three this leads to a variation on a recent theorem of Capovilla, Löh and Moraschini [1]. [ABSTRACT FROM AUTHOR]

Abstract: Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of - and -bundles over . In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori. [Copyright &y& Elsevier]

We extend the product formula for Lusternik-Schnirelmann category and topological complexity to some classes of fiber bundles. Then we compute and study the Lusternik-Schnirelmann category and topological complexity of Dold manifolds. [ABSTRACT FROM AUTHOR]

Abstract: Instead of Lusternik–Schnirelmann -category we define the -cohomological dimension by the cohomology with local coefficients and a gap theorem for closed manifolds is shown to hold for this invariant. [Copyright &y& Elsevier]

We prove that, for every decreasing sequence {ak} of natural numbers, there exists a map f :X→X with catfk=ak. [Copyright &y& Elsevier]