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 complex. Even stronger, the main result states that, if $C_*$ is a free chain complex, and \cm an acyclic matching, then C_*=C_*^\cm\oplus T_*, where C_*^\cm is the Morse complex generated by the critical elements, and $T_*$ is an acyclic complex

Discrete Morse theory for computing cellular sheaf cohomology

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.

Algebraic Morse theory and homological perturbation theory

We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim