Odd-Cycle-Free Facet Complexes and the K\"onig property

We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the K\"onig property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the K\"onig property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.Comment: 12 pages, 11 figure

Eilenberg swindles and higher large scale homology of products of trees

We show that uniformly finite homology of products of $n$ trees vanishes in all degrees except degree $n$, where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of trees. We also show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.Comment: Final version, to appear in Groups, Geometry & Dynamic

Cubical Cohomology Ring of 3D Photographs

Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological invariants in the image context. In this paper, we present formulas to directly compute the cohomology ring of 3D cubical complexes without making use of any additional triangulation. Starting from a cubical complex $Q$ that represents a 3D binary-valued digital picture whose foreground has one connected component, we compute first the cohomological information on the boundary of the object, $\partial Q$ by an incremental technique; then, using a face reduction algorithm, we compute it on the whole object; finally, applying the mentioned formulas, the cohomology ring is computed from such information

A good leaf order on simplicial trees

Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of H\`a and Van Tuyl for computing the graded Betti numbers of simplicial trees.Comment: 17 pages, to appear; Connections Between Algebra and Geometry, Birkhauser volume (2013