In this paper we will present a very simple discrete Morse theory for CW complexes. 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 gradient flow associated to a Morse function. Using this, 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.
This Morse theory takes on particular significance in the context of PL manifolds. To clarify this statement, we take a small historical digression. In 1961, Smale proved the h-cobordism theorem for smooth manifolds (and hence its corollary, the Poincare conjecture in dimension 5) using a combination of handlebody theory and Morse theory . In [Mi2], Milnor presented a completely Morse theoretic proof of the h-cobordism theorem. In ([Ma], [Ba], [St]) Mazur, Barden and Stallings generalized Smale's theorem by replacing Smale's hypothesis that the cobordism be simply-connected by a weaker simple-homotopy condition. Along the way, this more general theorem (the s-cobordism theorem) was extended to other categories of manifolds. In particular, a PL s-cobordism theorem was established. In this case, it was necessary to work completely within the context of handlebody theory. The Morse theory presented in this paper can be used to give a Morse theoretic proof of the PL s-cobordism theorem, along the lines of the proof in .
In the remainder of this introduction, we present an informal exposition of the contents of the paper. To avoid minor complications we will restrict attention, in this introduction, to simplicial complexes.
Let M be any finite simplicial complex, K the set of simplices of M, and K p the simplices of dimension p. A discrete Morse function on M will actually be a function on K. That is, we assign a single real number to each Article No. AI971650 90 0001-8708ร98 25.00