Lusternik Schnirelmann Category Research Articles

Multiple Normalized Solutions to a Logarithmic Schrödinger Equation via Lusternik–Schnirelmann Category

Multiple Normalized Solutions to a Logarithmic Schrödinger Equation via Lusternik–Schnirelmann Category

Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains

In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p,q)-Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at least γ(Ωλ∖{0}) pairs (±u) of odd weak solutions with precisely two nodal domains, where γ stands for the genus.

Topological Complexity and LS-Category of Certain Manifolds

The Lusternik–Schnirelmann category and topological complexity are important invariants of topological spaces. In this paper, we calculate the Lusternik–Schnirelmann category and topological complexity of products of real projective spaces and their wedge products by using cup and zero-cup length. Also, we will find the topological complexity of R P 2 k + 1 by using the immersion dimension of R P 2 k + 1 .

Multiplicity of solutions to a nonlinear elliptic problem on a Riemannian orbifold

We employ the photography method to estimate the number of solutions to a nonlinear elliptic problem on a Riemannian orbifold in function of the Lusternik–Schnirelmann category of its submanifold of points with largest local group.

Different types of topological complexity based on higher homotopic distance

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.

The p-widths of a surface

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the p-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets.We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the p-widths of the round sphere are attained by lfloor sqrt{p}rfloor great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be sqrt{pi }.En route to calculating the p-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.

On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole \(\mathbb{R}^N\) via Lusternik-Schnirelmann Category

On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole \(\mathbb{R}^N\) via Lusternik-Schnirelmann Category

G-category versus orbifold category

We present a comparative study of certain invariants defined for group actions and their analogues defined for orbifolds. In particular, we prove that Fadell's equivariant category for $G$-spaces coincides with the Lusternik-Schnirelmann category for orbifolds when the group is finite.

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy–Littlewood–Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

Normal forms, invariant manifolds and Lyapunov theorems

<abstract><p>We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.</p></abstract>

Equivariant dimensions of groups with operators

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension $\mathsf{cd}\_G(\pi)$, the equivariant geometric dimension $\mathsf{cat}\_G(\pi)$, and the equivariant Lusternik–Schnirelmann category $\mathsf{gd}\_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg–Ganea and Stallings–Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with $\mathsf{cat}\_G(\pi)=\mathsf{cd}\_G(\pi)=2$ and $\mathsf{gd}\_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings–Swan type result for families of subgroups which do not contain all finite subgroups.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

Estimates of covering type and minimal triangulations based on category weight

Abstract In a recent publication [D. Govc, W. A. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 2020, 1, 31–48], we have introduced a new method, based on the Lusternik–Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this current paper, we present an important extension that takes into account the fundamental group of X. In fact, if π 1 ⁢ ( X ) {\pi_{1}(X)} contains elements of finite order, then one can often find cohomology classes of high ‘category weight’, which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.

On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds

On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds

Rational cohomology of classifying spaces and Hilali conjecture

Let X be a simply connected space with finite-dimensional rational homotopy groups. Denote by Bau 1 (X) the classifying space of fibrations with fiber X and max π ∗ (X)=max {i; π i (X)≠0}. In this paper, we show that the dimensional rational cohomology and the Lusternik-Schnirelmann category of Bau 1 (X) are infinite if max π ∗ (X) is odd. Our results apply, in particular, when X is elliptic. As a consequence, we prove the Hilali conjecture for classifying space.

Homotopy Theory in Digital Topology

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

On the LS-category and topological complexity of projective product spaces

We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtain 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.

On the topological complexity of manifolds with abelian fundamental group

Abstract We find conditions which ensure that the topological complexity of a closed manifold M with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik–Schnirelmann category of the cofibre of the diagonal map Δ : M → M × M {\Delta:M\to M\times M} for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds.

A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

Nonlinear spectrums of Finsler manifolds

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler–Laplacian operator, we introduce faithful dimension pairs by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik–Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry–Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik–Schnirelmann dimension pair.