Discrete Morse Theory Is At Least As Perfect As Morse Theory
Abstract
In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show that some subdivision of the triangulation admits a boundary-critical discrete Morse function with c_i interior critical faces of dimension d-i. This dualizes and extends a recent result by Gallais. Further consequences of our work are: (1) Every simply connected smooth d-manifolds (except possibly when d=4) admits a locally constructible triangulation. (This solves a problem by Zivaljevic.) (2) Up to refining the subdivision, the classical notion of geometric connectivity can be translated combinatorially via the notion of collapse depth.
Keywords
Cite
@article{arxiv.1010.0548,
title = {Discrete Morse Theory Is At Least As Perfect As Morse Theory},
author = {Bruno Benedetti},
journal= {arXiv preprint arXiv:1010.0548},
year = {2014}
}
Comments
This paper will be subsumed by (and merged into) a new version of arXiv:1212.0885, to appear soon