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A user’s guide to discrete Morse theory. (English) Zbl 1048.57015
A discrete Morse function \(f\) assigns real values to the faces of a CW-complex. Such a function is required to generally increase with the dimension of the faces, but for each \(k\)-face \(\sigma\) there is at most one exception allowed: there may be either one \((k+1)\)-face containing \(\sigma\) with a value lower than \(f(\sigma)\) or one \((k-1)\)-face contained in \(\sigma\) with a value higher than \(f(\sigma)\). The faces which do not make use of this exception are called critical. Based on these elementary notions, the author developed a combinatorial adaptation of smooth Morse theory in [Adv. Math. 134, No. 1, 90–145 (1998; Zbl 0896.57023)]. Since then this has proved to be useful in solving a variety of problems in combinatorial topology and related areas. The paper under review surveys some of these results along with a very readable introduction to the theory.

57Q05 General topology of complexes
57R70 Critical points and critical submanifolds in differential topology
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