Since the Euler characteristic of P is inherently related to both the combinatorial and topological structure of P, we will also be interested in studying the (reduced) Betti numbers of P (over a field k), which are defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
3 is sufficient to characterize the h'-numbers of Buchsbaum simplicial posets with prescribed Betti numbers.
j] satisfying the conditions of the theorem, a Buchsbaum simplicial poset having those Betti numbers and h'-numbers.
He begins with manifolds, tensors and exterior forms and progresses to such topics as the integration of differential forms and the Lie derivative, the Poincare Lemma and potentials, Monkowski space, covariant differentiation and curvature, relativity, Betti numbers
and De Rham's theorem, harmonic forms, the Aharonov-Bohm effect, and Yang- Mills fields.
These invariants include the minimal number of generators, deficiency, Betti numbers
over arbitrary fields, various spectral and representation theoretic invariants, graph polynomials and entropy.
Computing Betti numbers via combinatorial Laplacians.
Equivalently, the vectors in Cut([SIGMA]) support sets of facets whose deletion increases the codimension-1 Betti number, and the vectors in Flow([SIGMA]) support nontrivial rational homology classes.
a) Enumerative criterion: We give a simple criterion for the r-stackedness in terms of h-vectors and Betti numbers for homology manifolds with boundary (Theorem 3.
b) Vanishing of Betti numbers and missing faces: We show that if a homology manifold (with or without boundary) is r-stacked, then it has zero Betti numbers and no missing faces in certain dimensions (Corollary 3.
If one knows the Betti numbers of [DELTA], then knowing h([DELTA]) is equivalent to knowing h'([DELTA]) (or h"([DELTA])).
When G is a complete graph, the syzygies and Betti numbers of the ideal in([I.
The description of the generating sets and the Betti numbers is in terms of the "connected flags" of G.