Betti Numbers Of Simplicial Complexes Research Articles

Extremal problems related to Betti numbers of flag complexes

We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32n for the sum of Betti numbers of any n-vertex flag complex and 1.25n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations.

Nerve complexes and moment-angle spaces of convex polytopes

We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope P to the nerve KP of the cover of the boundary ∂P by facets. We develop a theory of nerve complexes and apply it to the moment-angle spaces ZP of convex polytopes P. In the case of a polytope P with m facets, its moment-angle space ZP is defined by the canonical embedding in the cone ℝ≥m. It is well-known that the space ZP is homeomorphic to the polyhedral product (D2, S1)∂P* if the polytope P is simple. We show that the homotopy equivalence \(\mathcal{Z}_P \simeq (D^2 ,S^1 )^{K_P }\) holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known f-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, f-polynomials) coincide, while the new invariants are different.

Computing Betti Numbers via Combinatorial Laplacians

We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ν , of eigenvalues which we have yet to understand fully. We numerically verify a conjecture of Bj{o}rner, Lov{a}sz, Vre{\'c}ica, and {\u Z}ivaljevi{\'c} on the chessboard complexes C(4,6) , C(5,7) , and C(5,8) . Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C(4,6) and C(5,8) , giving a completely rigorous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.

An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere

A general and direct method for computing the Betti numbers of a finite simplicial complex in S d is given. This method is complete for d ⩽ 3, where versions of this method run in time O( nα( n)) and O( n), n the number of simplices. An implementation of the algorithm is applied to alpha shapes, which is a novel geometric modeling tool.