We prove an analogue of the Approximation Theorem of L-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.

The recent proof of the Boij-Söderberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup of Betti diagrams. We prove that this semigroup is finitely generated, and we answer several other fundamental questions about this semigroup.

Betti numbers can be used as a means for feature detection to aid in the exploration of complex large-scale data sets. We present a fast algorithm for the calculation of Betti numbers for triangulated isosurfaces, along with examples of their use. Once an isosurface is extracted from a data set, calculating Betti numbers only requires time and space proportional to the isosurfaces, not the data...

We find strong necessary conditions on the f -vectors, Betti sequences, and relative Betti sequence of a pair of simplicial complexes. We also present an example showing that these conditions are not sufficient. If only the difference between two Betti sequences is specified, and not the individual Betti sequences, then the characterization is complete, and the characterization of all pairs of ...

Let S = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let ∆ be a simplicial complex on [n] = {1, . . . , n} and I∆ ⊂ S its Stanley–Reisner ideal. We write ∆e for the exterior algebraic shifted complex of ∆ and ∆c for a combinatorial shifted complex of ∆. Let βii+j(I∆) = dimK Tori(K, I∆)i+j denote the graded Betti numbers of I∆. In the present p...

Let P be a finite partially ordered set with unique minimal element 0̂. We study the Betti poset of P , created by deleting elements q ∈ P for which the open interval (0̂, q) is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form (0̂, p) ⊂ P and corresponding open intervals in the Betti poset. Our motivating application is that the...

We consider the open problem of determining the graded Betti numbers for fat point subschemes Z supported at general points of P. We relate this problem to the open geometric problem of determining the splitting type of the pullback of ΩP2 to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one fo...

In note' I made use of the idea of the product of two constructs in i. ,obtain certain properties of Abelian integrals attached to algebra.. iieties. The present note uses the same method to, obtain further pi perties of such integrals. §1. 1. Let A, B be two homeomorphic absolute manifolds of n dimensions, and consider the product A X B. On this there is a cycle r of n dimensions, homeomorphic...

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vrećica, and Z̆ival...

Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti