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In this paper, we investigate vector fields on polyhedral complexes and theirassociated trajectories. We study vector fields which are analogue of thegradient vector field of a function in the smooth case. Our goal is to define anice theory of trajectories of such vector fields, so that the set of themcaptures the topology of the polyhedral complex, as in classical Morse theory.Since we do not assume the polyhedral complex to be a manifold, the definitionof vector fields on it is very different from the smooth case. Nevertheless, wewill show that there exist nice classes of functions and metrics which givegradient vector fields with desired properties. Our construction relies onForman's discrete Morse theory. In particular, the class of functions we use isan improvement of Forman's discrete Morse functions. A notable feature of ourtheory is that our gradient vector fields are defined purely from functions andmetrics as in the smooth case, contrary to the case of discrete Morse theorywhere we need the data of dimension of cells. This allows us to implementseveral useful constructions which were not available in the discrete case.

Integration of vector fields on cell complexes and Morse theory