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