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A discrete Morse theory for cell complexes. (English) Zbl 0867.57018
Yau, S.-T. (ed.), Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott’s 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 112-125 (1995).
We present a very simple discrete Morse theory for cell complexes. This Morse theory first appeared in [the author, Morse theory for cell complexes (preprint)]. In addition to proving analogues of the main theorems of Morse theory, we also present discrete analogues of such (seemingly) intrinsically smooth notions as the gradient vector field and the corresponding gradient flow associated to a Morse function. Using these, we define a Morse complex, a differential complex built out of the critical points of our discrete Morse function, which has the same homology as the underlying manifold. We conclude by presenting discrete analogues of E. Witten’s Hodge theoretic approach to Morse theory [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)]. Most of this paper will be an informal exposition of the contents of [the author, Morse theory for cell complexes (preprint)] and [the author, Witten-Morse theory for cell complexes (preprint)].
For the entire collection see [Zbl 0826.00038].

57Q99 PL-topology