We construct a spectral sequence for L-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundame...

The notion of a characteristic fibration is introduced. This fibration consists of a base space M and a set of fibres which are dimension groups associated to a noncommutative ring R. Every dimension group of the fibration is isomorphic to the first Betti group of M with a ‘positive cone’ depending continuously on the fibre. The characteristic fibrations are linked to the codimension–1 regular ...

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...

Main theorems of the article concern the problem of M. Atiyah on possible values of l2-Betti numbers. It is shown that all non-negative real numbers are l2-Betti numbers, and that “many” (for example all non-negative algebraic) real numbers are l2-Betti numbers of aspherical manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to an aspherical m...

Recall that the (Mumford-Castelnuovo) regularity of M is the least integer ρ such that for each i all free generators of Fi lie in degree ≤ i + ρ, that is βi,j = 0, for j > i + ρ. In terms of Macaulay [Mac] regularity is the number of rows in the diagram produced by the “betti” command. A Betti number βi,j 6= 0 will be called extremal if βl,r = 0 for all l ≥ i and r ≥ j + 1, that is if βi,j is ...

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: beta{0} is the number of connected components and beta{k} effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously bee...

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2× 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre a...

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij–Söderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a...

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n t...

In recent work, Clark and Tchernev introduced the notion of monomial resolution supported on a poset. In this talk, I will review this notion, and the related notion of monomial resolution supported on a CW complex. I will discuss a result of mine that shows that resolutions supported on a CW complex are also supported on the face poset of a CW complex. This relates to the work of Clark and Tch...