On Quillen's Theorem A for posets

A theorem of McCord of 1966 and Quillen's Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads then to stronger formulations of the simplicial version of Quillen's Theorem A, the Nerve lemma and other known results.Comment: 7 pages

Strong d-collapsibility

We introduce a notion of strong d-collapsibility. Using this notion, we simplify the proof of Matoušek and the author showing that the nerve of a family of sets of size at most d is d-collapsible

Vertex decompositions of two-dimensional complexes and graphs

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug

Lusternik-Schnirelmann category of simplicial complexes and finite spaces

In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of category are obtained in the setting of finite topological spaces.Comment: 18 pages, 10 figures, this is a new version with some minor changes and a new exampl

A note on the homotopy type of the Alexander dual

We investigate the homotopy type of the Alexander dual of a simplicial complex. It is known that in general the homotopy type of K does not determine the homotopy type of its dual K∗ . We construct for each finitely presented group G, a simply connected simplicial complex K such that π1(K∗ ) = G and study sufficient conditions on K for K∗ to have the homotopy type of a sphere. We extend the simplicial Alexander duality to the more general context of reduced lattices and relate this construction with Bier spheres using deleted joins of lattices. Finally we introduce an alternative dual, in the context of reduced lattices, with the same homotopy type as the Alexander dual but smaller and simpler to compute.Fil: Minian, Elias Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaFil: Rodríguez, Jorge Tomás. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentin