Discrete Morse Theory for Manifolds with Boundary

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities” relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3 and for each k ≥ 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d− 1)-cells. (This solves a problem by Chari.) (2) For fixed d and k, there are exponentially many combinatorial types of simplicial d-manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d − 1)cells. (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.) (3) The barycentric subdivision of any simplicial constructible d-ball is collapsible. (This “almost” solves a problem by Hachimori.) (4) Every constructible ball collapses onto its boundary minus a facet. (This improves a result by the author and Ziegler.) (5) Any 3-ball with a knotted spanning edge cannot collapse onto its boundary minus a facet. (This strengthens a classical result by Bing and a recent result by the author and Ziegler.)