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:
For each and for each , there is a PL -sphere on which any discrete Morse function has more than critical -cells.
(This solves a problem by Chari.)
Focus Area 3: Topological connectivity and diameter of Discrete Structures
For fixed and , there are exponentially many combinatorial types of simplicial -manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most critical interior -cells.
(This connects discrete Morse theory to enumerative combinatorics/
discrete quantum gravity.)
The barycentric subdivision of any simplicial constructible -ball is
(This ``almost'' solves a problem by Hachimori.)
Every constructible ball collapses onto its boundary minus a facet.
(This improves a result by the author and Ziegler.)
Any -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.)