arXiv research

The paper proves a theorem linking convex body centroids and category theory.

We will provide a lower bound for the equivariant Lusternik-Schnirelmann category of an arbitrary proper action in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant Lusternik-Schnirelmann category of its generalized Weyl group. As an application we repro…

The study finds the minimum number of critical points for functionals on Frechet spaces and Finsler manifolds.

Firstly, we wish to motivate that Conley pairs, realized via Salamon's definition [17], are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces [21]. From this fell off quite naturally, firstly, an alternative proof [20] of the cell a…

The study proves the existence of geodesics on reversible Finsler spheres.

We define the LS-category cat_g 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 Dranishnikov's t…

We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids.

New bounds on shortest geodesic loops on a sphere.

We prove the Arnold conjecture for closed symplectic manifolds with $π_2(M)=0$ and $\cat M=\dim M$. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.

Formula proved for Lusternik-Schnirelmann category of connected sums of manifolds.

The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.

The minimal number of critical points is studied for smooth functions on closed manifolds.

This paper extends Lusternik-Schnirelmann category to non-compact manifolds.

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamenta…

Study probabilistic category and complexity bounds, comparing with classical invariants.

We extend Lusternik-Schnirelmann theory to pairs $(f, φ)$, where $φ$ is a homotopy equivalence of a space $X$, $f$ is a function on $X$ which decreases along $φ$ and $(f, φ)$ satisfies a discrete analog of the Palais-Smale condition. The theory is carried out in an equivariant setting.

Topological complexity for closed 1-forms

Surgery method proves category inequality for specific manifolds.

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient…

Study finds lower bounds for solutions on Riemannian orbifolds.

New coarse LS-category introduced for groups and spaces.

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + δh)$ has at least $Cat(M) +1$ solutions for $δ$ small enough, where $Cat(M)$ denotes the Lusternik-Schnirelmann-categ…

The paper proves rigidity theorems for area widths of Riemannian manifolds.

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, ω)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $π_2(M)=0$. The Arnold conjecture asserts that the number of…

The study confirms a conjecture about critical points of smooth functions.

Study numerical invariants under retraction maps between topological spaces.

The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik-Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant TC(X) measures the complexity of all possible mot…

The study finds multiple solutions for constant Q-curvature metrics.

Study proves existence of closed geodesics on spheres and projective spaces.

New TC variant dTC better fits motion planning for some systems.

New invariants help find closed geodesics on curved spaces.

Maps from 2-planes to projective spaces using quaternions and octonions.

The paper explores the topology and curvature of isoparametric families in spheres.

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with va…

We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we pa…

The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler-Laplacian operator, we introduce \textit{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 eigen…

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 $(M,ω)$.

New theorem connects distant points and identical points on manifolds.

We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map for non-orientable surfaces equals three. Also, we prove that the topological complexity of non-orientable surfaces of genus $>3$ is four.

We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and d…

In this paper, we study the spectrums of faithful dimension pairs on a closed Finsler manifold and obtain a Gromov type and a Buser type lower bounds for eigenvalues. Furthermore, for the Lusternik-Schnirelmann spectrum, we not only obtain a better lower bound, but also estimate the multiplicity of each eigenvalue.

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p>1$. We study critical points of geometric costs for these affine control systems, proving that if the base m…

Let $S$ be a set of critical points of a smooth real-valued function on a closed manifold $M$. Generalizing a well-known result of Lusternik--Schnirelmann, Reeken~[R] proved that $\cat S \geq \cat M$. Here we prove a generalization of Reeken"s inequality for gradient-like flows on compact spaces.

We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2. As an application, we prove that Lusternik-Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our…

Proves existence of multiple solutions to a multiphasic equation on manifolds.

Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\c…