Morse shellability, tilings and triangulations

25 pages, 9 figuresWe introduce notions of tilings and shellings on finite simplicial complexes, called Morse tilings and shellings, and relate them to the discrete Morse theory of Robin Forman.Skeletons and barycentric subdivisions of Morse tileable or shellable simplicial complexes are Morse tileable or shellable. Moreover, every closed manifold of dimension less than four has a Morse tiled triangulation, admitting compatible discrete Morse functions, while every triangulated closed surface is even Morse shellable. Morse tilings extend a notion of $h$-tilings that we introduced earlier and which provides a geometric interpretation of $h$-vectors. Morse shellability extends the classical notion of shellability