The Morse Theory of Cech and Delaunay Complexes
Ulrich Bauer, Herbert EdelsbrunnerDescription
Given a finite set of points in $\mathbb{R}^n$ and a radius parameter, the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexesin the light of generalized discrete Morse theory are investiagted. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, it is proven that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.
References
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Ulrich Bauer and Herbert Edelsbrunner.
The Morse theory of Čech and Delaunay complexes.
Transactions of the American Mathematical Society, 369(5):3741–3762, 2017.
arXiv:1312.1231, doi:10.1090/tran/6991.