Some remarks on Morse theory for posets, homological Morse theory and finite manifolds

We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Forman's discrete Morse theory for CW-complexes and generalizes Forman and Chari's results on the face posets of regular CW-complexes. We also introduce a homological variant of the theory that can be used to study the topology of triangulable homology manifolds by means of their order triangulations.Comment: In this revised version, the main results of the previous version have been generalized and improved. We also included various examples to illustrate the applications of this theor

A new approach to Whitehead's asphericity question

We investigate Whitehead's asphericity question from a new perspective, using results and techniques of the homotopy theory of finite topological spaces. We also introduce a method of reduction to investigate asphericity based on the interaction between the combinatorics and the topology of finite spaces.Comment: 9 pages, 5 figure

The Geometry of Relations

The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex ΔX of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84-95, 1952), we define the simplicial complexes KX and LX associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex ΔX. We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that KX and LX are topologically equivalent to the smaller complexes K′X, L′X induced by the relation ≤. More precisely, we prove that KX (resp. LX) simplicially collapses to K′X (resp. L′X). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y.Fil: Minian, Elias Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin