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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]

Motivated by the concept of S1-category of a manifold, we obtain a classification of all closed 3-manifolds that can be covered by three open solid tori. [ABSTRACT FROM AUTHOR]

We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well-defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the existence of positive solutions to the following Schrödinger–Poisson–Slater system where Ω is a bounded domain in , λ is a fixed positive parameter and . We prove that if p is “near” the critical Sobolev exponent , then the number of positive solutions is greater then the Lusternik–Schnirelmann category of Ω. [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]

It follows from a theorem of Gromov that the stable systolic category $${\rm cat}_{\rm stsys} M$$ of a closed manifold M is bounded from below by $${\rm cl}_{\mathbb{Q}} M$$, the rational cup-length of M [Ka07]. We study the inequality in the opposite direction. In particular, combining our results with Gromov’s theorem, we prove the equality $${\rm cat}_{\rm stsys} M = {\rm cl}_{\mathbb{Q}} M$$ for simply connected manifolds of dimension ≤ 7. [ABSTRACT FROM AUTHOR]

Let X, Y be finite 1-connected p-local CW-complexes. We show that cat(X × Y ) = cat(X) + cat(Y ) holds if p is large and the loop space homology of X, Y satisfies a certain flatness assumption. [ABSTRACT FROM AUTHOR]

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]

In this note we consider some functionals restricted to the nonholonomic constraint satisfied by lightlike curves in Lorentzian manifolds. These functionals are related to the arrival time map, whose critical points are light rays in general relativity, and they have been used in the variational theory of light rays in some recent literatures. We exhibit an explicit example showing that, from a functional point of view, the constraint above does not have sufficient regularity. Nevertheless, we show that up to a continuous change in coordinates, the used functionals restricted to the constraint have just the regularity that one needs to develop Lusternik–Schnirelmann theory or Morse theory. [ABSTRACT FROM AUTHOR]

The geometric category of a space was defined by R. H. Fox and it is a generalization of the Lusternik-Schnirelmann category. In the first part of this paper we give a characterization theorem of geometric category. Then we prove the product inequality and give some examples. In the last part of the paper we introduce the relative geometric category and we study some of its properties. [ABSTRACT FROM AUTHOR]

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation $mathcal {F}$ on a closed manifold $M$. The essential transverse category $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is introduced in this paper, and we prove that $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category $operatorname {cat}_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is finite, and thus $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F}) = operatorname {cat}_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ holds. par A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ into a standard problem about $mathbf O(q)$-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated $mathbf O(q)$-manifold $widehat W$, we have that $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F}) = operatorname {cat}_{mathbf O(q)}(widehat W)$. Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. par A generalization of the Lusternik-Schnirelmann theorem is derived: given a $C^1$-function $f colon M to mathbb {R}$ which is constant along the leaves of a Riemannian foliation $mathcal {F}$, the essential transverse category $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is a lower bound for the number of critical leaf closures of $f$. [ABSTRACT FROM AUTHOR]

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: In this paper, we investigate the existence of multiple solutions for critical inhomogeneous elliptic problem in non-contractible domains. We show that for satisfying a suitable condition and the Dirichlet problem admits at least four positive solutions, where , and are sufficiently small. [Copyright &y& Elsevier]

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]

The following inequality $$mathrm {cat}_{mathrm {LS}} Xle mathrm {cat}_{mathrm {LS}} Y+bigg lceil frac {hd(X)-r}{r+1}bigg rceil $$ holds for every locally trivial fibration $f:Xto Y$ between $ANE$ spaces which admits a section and has the $r$-connected fiber, where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that $$mathrm {cat}_{mathrm {LS}} Xle cd(pi _1(X))+bigg lceil frac {dim X-1}{2}bigg rceil $$ for every complex $X$ with $cd(pi _1(X))le 2$, where $cd(pi _1(X))$ denotes the cohomological dimension of the fundamental group of $X$. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the notion of an inductive category in a model category agrees with the Ganea approach given by Doeraene. This notion also coincides with the topological one when we consider the category of (well-)pointed topological spaces. An application is given. [Copyright &y& Elsevier]

We study the number of Darboux charts needed to cover a closed connected symplectic manifold (M, ω) and effectively estimate this number from below and from above in terms of the Lusternik–Schnirelmann category of M and the Gromov width of (M, ω). [ABSTRACT FROM AUTHOR]

Abstract: We show that the number of solutions of a nonlinear elliptic problem on a Riemannian manifold depends on the topological properties of the manifold. In particular we consider the Lusternik–Schnirelmann category and the Poincaré polynomial of the manifold. [Copyright &y& Elsevier]

Abstract: In this paper we obtain the existence of multiple nodal solutions for quasilinear elliptic equation with Hardy–Sobolev critical exponents of the type where Ω is a bounded domain with smooth boundary in and contains 0 in its interior, , , , , λ is a positive parameter, and . By Lusternik–Schnirelmann category theory, we show the effect of the domain topology on the number of minimal nodal solutions for . [Copyright &y& Elsevier]

We prove the existence of infinitely many stationary solutions of a nonlinear one-dimensional superlinear equation on time scales using a Lusternik-Schnirelmann type result. [ABSTRACT FROM AUTHOR]

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holdër continuous function on M. We prove multiplicity of changing sign solutions for equations like Δg u + hu = |u|2* - 2 u, where Δg is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint. [ABSTRACT FROM AUTHOR]

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]

A highly efficient lossless encoding method for static images is proposed. In this method, multiple linear predictors are created for each image and adaptive prediction that responds to the local structure of images such as edges and textures is achieved by switching between these predictors at the block level. Furthermore, the probability density functions of the prediction errors are categorized by context modeling and modeled by generalized Gaussian functions, and adaptive arithmetic encoding of the prediction errors is performed by using probability tables that are generated for each pixel from this model. Parameters that are needed in the coding such as the prediction coefficients, the predictor selection data for each block, and the shapes of the generalized Gaussian functions are optimized by repeatedly minimizing a cost function that includes the code length of the parameters themselves in addition to the code length of the prediction errors that are calculated from the probability model above, and the parameters are then encoded separately as side data for each image. A procedure is introduced to improve prediction accuracies by using quadtree segmentation to segment the image into variable-sized blocks between which the predictor can change. Coding experiments are conducted and the proposed method is found to produce coding rates of 6 to 44% lower than the international standard JPEG-LS method, with the proposed method achieving superior coding performance that surpasses existing coding methods for all of the images used in the experiments. © 2007 Wiley Periodicals, Inc. Syst Comp Jpn, 38(4): 1–11, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20630 [ABSTRACT FROM AUTHOR]

Let be a regular covering over a finite polyhedron with free abelian group of covering translations. Each nonzero cohomology class ξ ∈ H1( X; R) with q*ξ = 0 determines a notion of “ infinity” of the noncompact space . In this paper we characterize homology classes z in which can be realized in arbitrary small neighborhoods of infinity in . This problem was motivated by applications in the theory of critical points of closed 1-forms initiated in [Farber M.: Zeros of closed 1-forms, homoclinic orbits and Lusternik-Schnirelman theory. Topol. Methods Nonlinear Anal. 19 (2002), 123–152], [Farber M.: Lusternik-Schnirelman theory and dynamics. Lusternik-Schnirelmann Category and Related Topics. Contemporary Mathematics 316 (2002), 95–111]. [ABSTRACT FROM AUTHOR]

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]

Let φt be a continuous flow on a metric space X and I an isolated invariant set with an index pair (N, L) and a Morse decomposition {Mi}i=1n. For every category ν on N/L, we prove that ν(N/L) ≤ ν([L])+∑i=1n ν(Mi). As a result if φt∣I is gradient-like and X is semi-locally contractible, then φt has at least νH(h(I)) - 1 rest points in I, where h(I) is the Conley index of I and νH is the Homotopy Lusternik-Schnirelmann category. [ABSTRACT FROM AUTHOR]

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]

This paper is concerned with well-known Lusternik-Schnirelmann categories. We desire to find some links and relations among them. This has been done by using the concepts of precategoty, categorical collection and closure of a category. [ABSTRACT FROM AUTHOR]

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]

We provide new upper and lower bounds for the rational LS-category of a rational fibration $\xi :F\to E \to K(\mathbf{Q},2n)$ of simply connected spaces that depend on a measure of the triviality of $\xi $ which is strictly finer than the vanishing of the higher holonomy actions. In particular, we prove that if $\xi $ is {\em $k$-trivial} for some $k\ge 0$ and $H^{*}(F)$ enjoys Poincaré duality, then \begin{equation*}\operatorname{cat}_{0}E \ge \operatorname{cat}_{0}F +k.\end{equation*} [ABSTRACT FROM AUTHOR]

Algebraic approximations have proved to be very useful in the investigation of Lusternik–Schnirelmann category. In this paper the L.-S. category and its approximations are studied from the point of view of abstract homotopy theory. We introduce three notions of L.-S. category for monoidal cofibration categories, i.e., cofibration categories with a suitably incorporated tensor product. We study the fundamental properties of the abstract invariants and discuss, in particular, their behaviour with respect to cone attachments and products. Besides the topological L.-S. category the abstract concepts cover classical algebraic approximations of the L.-S. category such as the Toomer invariant, rational category, and the A- and M-categories of Halperin and Lemaire. We also use the abstract theory to introduce a new algebraic approximation of L.-S. category. This invariant which we denote by ℓ is the first algebraic approximation of the L.-S. category which is not necessarily ⩽1 for spaces having the same Adams–Hilton model as a wedge of spheres. For a space X the number ℓ(X) can be determined from an Anick model of X. Thanks to the general theory one knows a priori that ℓ is a lower bound of the L.-S. category which satisfies the usual product inequality and increases by at most 1 when a cone is attached to a space. [Copyright &y& Elsevier]

We determine the Lusternik–Schnirelmann (L–S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea''s Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L–S category as its ‘once punctured submanifold’ N&z.sbs;{P}, P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea''s conjecture are described. [Copyright &y& Elsevier]

This paper presents a new method for using cup product information to draw conclusions about the Lusternik–Schnirelmann category of a space. The key idea is that of the Hopf set in X of a map f : Sn−1→L; if K=L∪fDn⊆X, then catX(K)=catX(L) if and only if * is in the Hopf set in X of f. The main result explicitly constructs elements of the Hopf set in X of f in terms of members of the Hopf set in X of the attaching maps of lower dimensional cells. Applications include: a calculation of the category of Sp(2) without higher order cohomology operations; new, easily used upper bounds for Lusternik–Schnirelmann category that apply to any space; and new information about the category of the CW skeleta of loop spaces and free loop spaces on even-dimensional spheres. [Copyright &y& Elsevier]

Soit $X$ un espace ayant le type d'homotopie rationnelle d'un produit de sphères impaires. Si, pour tout nombre premier $p$, la LS-catégorie de tous les $p$-localisés de $X$ est majorée par $n$, nous montrons que la LS-catégorie de $X$ est majorée par $n+1$. Si $Y$ est un élément dans le genre de Mislin de $X$, nous en déduisons: $|{\operatorname{cat}} (Y) - {\operatorname{cat}}(X) |\leq 1$. Dans le cas d'un $H$-espace $X$ de rang~2, nous avons exactement ${\operatorname{cat}} (X) = {\operatorname{cat}}(Y)$, pour tout espace $Y$ dans le genre de $X$. [ABSTRACT FROM AUTHOR]

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]

This survey of model categories and their applications in algebraic topology is intended as an introduction for non homotopy theorists, in particular category theorists and categorical topologists. We begin by defining model categories and the homotopy-like equivalence relation on their morphisms. We then explore the question of compatibility between monoidal and model structures on a category. We conclude with a presentation of the Sullivan minimal model of rational homotopy theory, including its application to the study of Lusternik–Schnirelmann category. [ABSTRACT FROM AUTHOR]