Curvature and relative Betti numbers.

-betti Numbers

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 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 positive 1-relator g...

Coverings and Betti Numbers

Curvature, Diameter and Betti Umbers

Let V denote a compact (without boundary) connected Riemannian manifold of dimension n. We denote by K the sectional curvature of V and we set inf K = ird~ K(-r) where 9 runs over all tangent 2-planes in V. One calls V a manifold of non-negative curvature ff inf K>~0. This condition has the following geometric meaning. A n n-dimensional Riemannian manifold has non-negative curvature iff for eac...

Betti numbers of subgraphs

Let G be a simple graph on n vertices. LetH be either the complete graph Km or the complete bipartite graph Kr,s on a subset of the vertices in G. We show that G contains H as a subgraph if and only if βi,α(H) ≤ βi,α(G) for all i ≥ 0 and α ∈ Z. In fact, it suffices to consider only the first syzygy module. In particular, we prove that β1,α(H) ≤ β1,α(G) for all α ∈ Z if and only if G contains a ...

Betti Numbers and Injectivity Radii

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...