In the first draft of the program for this Summer School in Marrakech, we planned to expose some standard material in differential topology, and some recent developments in low dimensional topology. We had then a discussion with the organizers, and it appeared that a unifying view on the program of this School was to establish bridges between differential matter on one hand, and discrete or computational geometry on the other hand. So we decided to focus on combinatorial topology and include an introduction to Robin Forman discrete Morse theory. In this chapter, we start with some classical combinatorial topology, including piecewise linear manifolds. Then we define discrete Morse functions on CW-complexes and show that we get the usual theorems of Morse theory. Everything here should sound rather familiar for people knowing classical theory. However the relation between differentiable objects, such that gradient vector fields, and their discrete counterpart has still to be explored. The generic questions below could give starting points for interesting research.
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