Connecting Discrete Morse Functions via Birth-Death Transitions

We study transformations between discrete Morse functions on a finite simplicial complex via birth-death transitions--elementary chain maps between discrete Morse complexes that either create or cancel pairs of critical simplices. We prove that any two discrete Morse functions f₁, f₂ on a finite simplicial complex K are linked by a finite sequence of such transitions. As applications, we present alternative proofs of several of Forman's fundamental results in discrete Morse theory and study the topology of the space of discrete Morse functions.