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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Morse theory of Čech and Delaunay complexes
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by Ulrich Bauer and Herbert Edelsbrunner PDF
Trans. Amer. Math. Soc. 369 (2017), 3741-3762 Request permission

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
  • MR Author ID: 898087
  • 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
  • MR Author ID: 61770
  • Email: edels@ist.ac.at
  • 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
  • © Copyright 2016 American Mathematical Society
  • 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
  • MathSciNet review: 3605986