2017/04/19 by J. F. Peters, Peters, J. F.
Computer Science · Mathematics · Neuroscience · #54E05 #68U05 #Axon Guidance and Neuronal Signaling #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1704.05727
openalex publication_date 2017/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article introduces proximal Cech nerves and Cech complexes, restricted to finite, bounded regions K of the Euclidean plane. A Cech nerve is a collection of intersecting balls. A Cech complex is a collection of nerves that cover K. Cech nerves are proximal, provided the nerves are close to each other, either spatially or descriptively. A Cech nerve has an advantage over the usual Alexandroff nerve, since we need only identify the center and fixed radius of each ball in a Cech nerve instead of identifying the three vertices of intersecting filled triangles (2-simplexes) in an Alexandroff nerve. As a result, Cech nerves more easily cover K and facilitate approximation of the shapes of irregular finite, bounded planar regions. A main result of this article is an extension of the Edelsbrunner-Harer Nerve Theorem for descriptive and non-descriptive Cech nerves and Cech complexes, covering K.