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A Non-expensive Homological Helly Theorem

Affiliation:	1. Universidad Nacional de Rosario, Argentina;2. CONICET and Universidad Nacional de Rosario, Argentina;1. Universidad Nacional de Rosario, Argentina;2. Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina;1. CNRS and LIP6, Université Pierre et Marie Curie, Paris, France;2. Departamento Ingeniería Industrial, Universidad de Chile;1. Department of Computer Science, Technion, Haifa, Israel;2. Mathematical Institute, University of Oxford, Oxford, United Kingdom;3. Department of Computer Science, Tufts University, Medford, MA, USA;4. California State University, Northridge, CA, USA;5. University of Calgary, Calgary, AB, Canada

Abstract:	We will discuss the following result: for a topological space X with the property that $H_{k} (U) = 0$ for $k ⩾ d$ and every open subset U of X, a finite family of open sets in X has nonempty intersection if for any subfamily of size j, $1 ⩽ j ⩽ d + 1$ , the ( $d - j$ )-dimensional reduced homology group of its intersection is zero. We also use this theorem to discuss new results concerning transversal affine planes to families of convex sets.

Keywords:	Helly property transversal affine planes homology
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