Related papers: Transverse LS-Category for Riemannian Foliations

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

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound on the number of critical points of a smooth function on a manifold. Another similar invariant is the topological complexity $TC(X)$ (a la…

An upper bound for the $L^2$- norm of the Euler class $e(\cal F)$ of an arbitrary transversally orientable foliation $\cal F$ of codimension one, defined on a three-dimensional closed irreducible orientable Riemannian 3-manifold $M^3$ is…

We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary…

We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity…

We give a solution to the Poincar\'e Problem, in the formulation of Cerveau and Lins Neto. We obtain a bound on the degree of general leaves of foliations of general type, which is linear in $g$. To achieve this we study the birational…

We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the…

Inverse categories are categories in which every morphism x has a unique pseudo-inverse y in the sense that xyx=x and yxy=y. Persistence modules from topological data analysis and similarly decomposable category representations factor…

Let~$S^{n-1}\rightarrow E \rightarrow M^n$ be an oriented sphere bundle supporting an affine transverse foliation. We give an upper bound for the Euler number of the bundle. We also give a new and elementary proof of the following fact: if…

Recently, a new homotopy invariant of metric spaces, called the distributional LS-category, was defined, which provides a lower bound to the classical LS-category. In this paper, we obtain several sufficient conditions for the…

For any compact oriented manifold $M$, we show that that the top degree multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of…

If C=C_\phi denotes the mapping cone of an essential phantom map \phi from the suspension of the Eilenberg-Mac Lane complex K=K(Z,5) to the 4-sphere S=S^4 we derive the following properties: (1) The LS category of the product of C with any…

For a Riemannian foliation $F$ on a compact manifold $M$ with a bundle-like metric, the de Rham complex of $M$ is $C^{\infty}$-splitted as the direct sum of the basic complex and its orthogonal complement. Then the basic component…

Let $F \hookrightarrow X \to B$ be a fibre bundle with structure group $G$, where $B$ is $(d{-}1)$-connected and of finite dimension, $d \geq 1$. We prove that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim B}{d}$,…

For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics…

We develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants,…

The purpose of this paper is to both survey and offer some new results on the non-triviality of the characteristic classes of Riemannian foliations. We give examples where the primary Pontrjagin classes are all linearly independent. The…

We call every complex connected (1,1)-dimensional supermanifold a super Riemann surface and construct versal super families of compact ones, where the base spaces are allowed to be certain ringed spaces including all complex supermanifolds.…

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…