# zbMATH — the first resource for mathematics

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.

##### MSC:
 57Q05 General topology of complexes 57R70 Critical points and critical submanifolds in differential topology
Full Text: