Simplicial and Cubical Complexes: Analogies and Differences

The research summarized in this thesis consists essentially of two parts. In the first, we generalize a coloring theorem of Baxter about triangulations of the plane (originally used to prove combinatorially Brouwer's fixed point theorem in two dimensions) to arbitrary dimensions and to oriented simplicial and cubical pseudomanifolds. We show that in a certain sense no other generalizations may be found. Using our coloring theorems we develop a purely combinatorial approach to cubical homology. (This part is joint work with Richard Ehrenborg.) In the second part, we investigate the properties of the Stanley ring of cubical complexes, a cubical analogue of the Stanley-Reisner ring of simplicial complexes. We compute its Hilbert-series in terms of the f-vector. We prove that by taking the initial ideal of the defining relations, with respect to the reverse lexicographic order, we obtain the defining relations of the Stanley-Reisner ring of the triangulation via "pulling the vertices" of the cubical complex. We show that the Stanley ring of a cubical complex is Cohen-Macaulay when the complex is shellable and its defining ideal is generated by homogeneous forms of degree two when the complex is also a subcomplex of the boundary complex of a convex cubical polytope. We present a cubical analogue of balanced CohenMacaulay simplicial complexes: the class of edge-orientable shellable cubical complexes. Using Stanley's results about balanced Cohen-Macaulay simplicial complexes and the degree two homogeneous generating system of the defining ideal, we obtain an infinite set of examples for a conjecture of Eisenbud, Green and Harris. This conjecture says that the h-vector of a polynomial ring in n variables modulo an ideal which has an nelement homogeneous system of parameters of degree two, is the f-vector of a simplicial complex. Thesis Supervisor: Richard P. Stanley Title: Professor of Mathematics