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...

Certain necessary conditions on the face numbers and Betti numbers of simplicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Cohen-Macaulay complexes originally due to Stanley...

We introduce a notion of L-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of L-Betti numbers of countable discrete groups for lattices. In this way, several new computations are obtained for countable groups, including lattices in algebraic groups over local elds, and Kac-Moody lattices. We also extend the vanishing of reduce...

Let Z be a finite set of double points in P 1 × P 1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of X, for the bigraded Betti numbers of I Z , the defining ideal of Z. We then relate the total Betti numbers of I Z to the shifts in the graded resolution, thus answering a special ...

We generalize Lück’s Theorem to show that the L-Betti numbers of a residually amenable covering space are the limit of the L-Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L torsion is a homotopy invariant for such spaces. We give examples of residually amenable gro...

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions are known, i.e., for modules of codimension two, for Gorenstein algebras of codimension t...

We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with m non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times m simplices. Together with the known reduction in the other direction, this implies that the two problems have ...

Let (X,x) be a germ of an isolated canonical 3-fold singularity. Fix a representative X of the germ which is Stein and contractible. Let φ′ : Y ′ → X be a crepant projective morphism from Y ′ with terminal Q-factorial singularities, and ψ : Y → Y ′ an analytic Q-factorialisation of Y ′ at its singular points. The aim of this paper is to calculate the Betti numbers of Y or, equivalently, the int...

In [DJL07] it was shown that if A is an affine hyperplane arrangement in Cn, then at most one of the L2–Betti numbers b i (C n \ A, id) is non–zero. In this note we prove an analogous statement for complements of complex affine hypersurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of [FLM09, LM06] about L2–Betti numbers of pl...