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The Morse theory of Čech and Delaunay complexes


Authors: Ulrich Bauer and Herbert Edelsbrunner
Journal: Trans. Amer. Math. Soc. 369 (2017), 3741-3762
MSC (2010): Primary 52C99; Secondary 51F99, 55U10, 57Q10
DOI: https://doi.org/10.1090/tran/6991
Published electronically: December 27, 2016
MathSciNet review: 3605986
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Abstract: Given a finite set of points in $ \mathbb{R}^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.


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Additional Information

Ulrich Bauer
Affiliation: Department of Mathematics, Technical University of Munich, 85748 Garching, Germany
Email: ulrich-bauer.org

Herbert Edelsbrunner
Affiliation: Department of Computer Science, University of Illinois, Urbana, Illinois 61801
Address at time of publication: IST Austria, 3400 Klosterneuburg, Austria
Email: edels@ist.ac.at

DOI: https://doi.org/10.1090/tran/6991
Received by editor(s): August 11, 2015
Received by editor(s) in revised form: April 15, 2016, and May 31, 2016
Published electronically: December 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society