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.

This paper will be subsumed by (and merged into) a new version of arXiv:1212.0885, to appear soon

References in corpus (4)

• Completion of the Proof of the Geometrization Conjecture
• Nash homotopy spheres are standard
• Exotic smooth structures on $S^2\times S^2$
• Rank and genus of 3-manifolds

Cited by in corpus (2)

• Kauffman's clock lattice as a graph of perfect matchings: a formula for its height
• Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields