Discrete Morse Theory Is At Least As Perfect As Morse Theory

Details Discrete Morse Theory Is At Least As Perfect As Morse Theory Discrete Morse Theory Is At Least As Perfect As Morse Theory

2010-10-04

arxiv.org/abs/1010.0548v4

Mathematics

Differential Geometry

Revised version (17 pages)

Scientific paper

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.

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