Morse Theory for Cell Complexes

Discrete Morse Theory for free chain complexes

We extend the combinatorial Morse complex construction to the arbitrary free chain complexes, and give a short, self-contained, and elementary proof of the quasi-isomorphism between the original chain complex and its Morse

The Morse theory of \v{C}ech and Delaunay complexes

Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which a...

Extremal examples of collapsible complexes and random discrete Morse theory

We present extremal constructions connected with the property of simplicial collapsibility. (1) For each d ≥ 2, there are collapsible (and shellable) simplicial d-complexes with only one free face. Also, there are non-evasive d-complexes with only two free faces. (Both results are optimal in all dimensions.) (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contra...

Morse Theory

Discrete Morse Theory for Filtrations

Acknowledgements Enumerating all the ways in which I am grateful to Konstantin would essentially double the length of this document, so I'll save all that stuff for my autobiography. But I will note here that he was simultaneously patient, engaged, proactive, and – best of all – ruthlessly determined to refine and sculpt all our vague big-picture ideas into digestible and implementable concrete...