Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. We write ∆ for the exterior algebraic shifted complex of ∆ and ∆ for a combinatorial shifted complex of ∆. It will be proved that for all i and j one has the inequalities βii+j(I∆e) ≤ βii+j(I∆c) on the graded Betti numbers of I∆e and I∆c . In addition, the bad behavior of graded Betti numbers of I∆c will be studied.

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 . It is proved however, that the...

Let G be a chordal graph and I(G) its edge ideal. Let β(I(G)) = (β0, β1, . . . , βp) denote the Betti sequence of I(G), where βi stands for the ith total Betti number of I(G) and where p is the projective dimension of I(G). It will be shown that there exists a simplicial complex ∆ of dimension p whose f -vector f(∆) = (f0, f1, . . . , fp) coincides with β(I(G)).

Abstract. The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard grad...

We develop an iterated homology theory for simplicial complexes. This theory is a variation on one due to Kalai. For 1 a simplicial complex of dimension d − 1, and each r = 0, . . . , d , we define r th iterated homology groups of 1. When r = 0, this corresponds to ordinary homology. If 1 is a cone over 1′, then when r = 1, we get the homology of 1′. If a simplicial complex is (nonpure) shellab...

Analysis of multidimensional data is challenging. Topological invariants can be used to summarize essential features of such data sets. In this work, we propose to compute the Betti numbers from a generative model based on a simplicial complex learnt from the data. We compare it to the Witness Complex, a geometric technique based on nearest neighbors. Our results on different data distributions...

This paper presents statistical approaches to testing whether data are spherical in the topological sense, ie, they lie on a closed manifold with a hollow interior. We characterize the sphere by it’s Betti numbers and use the computational topology program PLEX [12] based on simplicial complexes to calculate the data’s Betti numbers. We use the parametric bootstrap approach to test the power of...

We study the persistent homology of random Čech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the kth persistent Betti number of a random Čech complex and determine its asymptotic order in the subcritical regime. This extends a result of...

We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each p, d, r, explicit bounds on the p-th Betti numbers of affine and projective subvarieties of Ck and PC, defined by r polynomials of degrees at most d as a function of p, d and r. Unlike previous bounds these bounds are independent of k, the dimension of the ambient s...