On the moduli space of discrete Morse functions

Abstract: In this work, we give a combinatorial-geometric model for moduli spaces of discrete Morse functions on CW complexes. We investigate properties of moduli spaces of discrete Morse functions and their relationship to the spaces of smooth Morse functions on manifolds, as well as spaces of discrete Morse matchings on CW complexes. Moreover, we provide a similar model for the moduli space of merge trees and realise the induced merge tree and the induced barcode as continuous maps between moduli spaces.