Discrete Morse Theory is at least as perfect as Morse Theory

Discrete Morse Theory is at least as perfect as Morse Theory

Bruno Benedetti – 2010

Focus Area 3: Topological connectivity and diameter of Discrete Structures
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 ci critical points of index i, we show that some subdivision of the triangulation admits
a boundary-critical discrete Morse function with ci
interior critical cells 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-manifold (d 6= 4) admits a locally constructible triangulation. (This solves a problem by Zivaljevi´c.) ˇ
(2) Up to reﬁning the subdivision, the classical notion of geometric connectivity can be translated
combinatorially via the notion of collapse depth

Title

Discrete Morse Theory is at least as perfect as Morse Theory