Mayer-Vietoris sequence for generating families in diffeological spaces

We prove a version of the Mayer-Vietoris sequence for De Rham differential forms in diffeological spaces. It is based on the notion of a generating family instead of that of a covering by open subsets.Comment: 12 pages. Version 2: we corrected several typos and introduced a new numberin

El mundo matemático digital: el proyecto WDML (World Digital Mathematics Library)

We describe the digitization process of the mathematical literature in the framework of the projects WDML and DML-E.Describimos el proceso de digitalización de la literatura matemática de investigación en el marco del proyecto WDML (World Digital Mathematics Library) y del proyecto español DML-E

Height functions on quaternionic Stiefel manifolds

In this note, we study height functions on quaternionic Stiefel manifolds and prove that all these height functions are Morse-Bott. Among them, we characterize the Morse functions and give a lower bound for their number of critical values. Relations with the Lusternik-Schnirelmann category are discussed

Homotopic distance and generalized motion planning

Attribution 4.0 International[Abstract]: We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifolds of any Morse–Bott function. This generalizes the Lusternik–Schnirelmann theorem (for Morse functions) and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.Ministerio de Economía, Industria y Competitividad ; MTM2016-78647-PXunta de Galicia ; ED431C 2019/10Ministerio de Ciencia, Innovación y Universidades ; FPU17/0344

Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem

We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik– Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott functions

Simplicial Lusternik-Schnirelmann category

The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms. We prove that it generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that it allows to develop many notions and results of algebraic topology which are costumary in the classical theory of Lusternik-Schnirelmann category. Also we compare the simplicial category of a complex with the LS-category of its geometric realization and we discuss the simplicial analogue of the Whitehead formulation of the LS-category