Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ its edge ideal. In this paper, we investigate the Betti numbers of via upper-Koszul simplicial complexes, splittings mapping cone construction. particular, provide recursive formulas for ideals several classes graphs. We also identify graphs whose have unique extremal number which allows us to compute regularity projective dime...

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...

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for ...

The theme of this paper is the connection between topological properties of a closed orientable hyperbolic 3-manifold M and the maximal injectivity radius of M . In [4] we showed that if the first Betti number of M is at least 3 then the maximal injectivity radius of M is at least log 3. By contrast, the best known lower bound for the maximal injectivity radius of M with no topological restrict...