Discrete Morse Theory and the Cohomology Ring

In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen and by Fukaya. 0. Introduction In [5] we introduced a discrete version of Morse theory, which could be applied to very general cell complexes. The reader can also see [4],[7] for introductory surveys, [6],[8] and [15] for extensions of the theory, and [9],[3],[1],[13], [14] and [16] for applications of the theory. In this paper we show how the ring structure of the cohomology of a simplicial complex can be seen from the point of view of discrete Morse theory. More generally, given any map on the cohomology ring which is induced by a cochain map, and any collection of discrete Morse functions, we give an explicit procedure for realizing the map by a map on the appropriate space of Morse cochains. Betz and Cohen [2], and, independently, Fukaya [11], [12], showed how one can see the ring structure of the cohomology of a smooth manifold from the point of view of standard Morse theory. Although we follow a slightly different point of view, the main ideas in this paper find their roots in these references. In particular, the idea of describing the relevant cochain maps in terms of graphs (see Figures 0.1–0.7 below) first appeared in these papers. Let us quickly review the basics of discrete Morse theory. The reader should consult [5] for a more complete presentation. Let M be a finite simplicial complex. For any simplex a of M , we will sometimes write a to indicate that a has dimension p. For simplices a and b of M , we will write a < b or b > a to indicate that a is a face of b. A function f : {simplices of M} → R Received by the editors August 13, 2001 and, in revised form, January 30, 2002. 2000 Mathematics Subject Classification. Primary 57Q99; Secondary 58E05. This work was partially supported by the National Science Foundation. c ©2002 American Mathematical Society