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Morse theory for cell complexes. (English) Zbl 0896.57023
This paper provides a discrete Morse theory for CW-complexes. The author proves an analogue of the main theorems of the classical theory and gets in the context of PL manifolds an interesting formula of the PL Poincaré conjecture.
“The following are equivalent: (1) (PL Poincaré conjecture) Let $$M$$ be a PL $$n$$-manifold which is a homotopy $$n$$-sphere. Then $$M$$ is a PL $$n$$-sphere; (2) Let $$M$$ be a PL $$n$$-manifold which is a homotopy $$n$$-sphere. Then, by a series of bisections, $$M$$ can be subdivided to a complex which has a Morse function with exactly 2 critical points.”
The author builds also the gradient vector field of a discrete (or combinatorial) Morse function and an associated differential complex with the same homology as the underlying manifold. This discrete Morse theory can then be used to give a Morse theoretic proof à la Milnor of the PL $$s$$-cobordism theorem [J. W. Milnor, Lectures on the $$h$$-cobordism theorem (1965; Zbl 0161.20302)].
Main of the final part of the paper is devoted to characterize gradient vector fields of discete Morse functions. Doing that, the author shows that every discrete Morse function can be replaced by a self-indexing one, with the same critical points.
In spite of some misprints or change without notice (e.g. combinatorial for discrete) this paper is pleasant and not too hard to read; there are many examples. It should become a reference in the subject.

MSC:
 57R70 Critical points and critical submanifolds in differential topology 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 57R80 $$h$$- and $$s$$-cobordism 57R60 Homotopy spheres, Poincaré conjecture 57Q99 PL-topology
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References:
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