Compact homogeneous spaces and the first Betti number.

Large Homogeneous Compact Spaces

A Class of compact operators on homogeneous spaces

Let  $varpi$ be a representation of the homogeneous space $G/H$, where $G$ be a locally compact group and  $H$ be a compact subgroup of $G$. For  an admissible wavelet $zeta$ for $varpi$  and $psi in L^p(G/H), 1leq p <infty$, we determine a class of bounded  compact operators  which are related to continuous wavelet transforms on homogeneous spaces and they are called localization operators.

The First L-Betti Number of Classifying Spaces for Variations of Hodge Structures

Classical Hodge theory gives a decomposition of the complex cohomology of a compact Kähler manifold M , which carries the standard Hodge structure {H(M), p + q = k} of weight k. Deformations of M then lead to variations of the Hodge structure. This is best understood when reformulating the Hodge decomposition in an abstract manner. Let HC = HR ⊕ C be a complex vector space with a real structure...

Betti number signatures of homogeneous Poisson point processes.

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...

The First L-betti Number and Approximation in Arbitrary Characteristic

Let G be a finitely generated group and G = G0 ⊇ G1 ⊇ G2 ⊇ · · · a descending chain of finite index normal subgroups of G. Given a field K, we consider the sequence b1(Gi;K) [G:Gi] of normalized first Betti numbers of Gi with coefficients in K, which we call a K-approximation for b (2) 1 (G), the first L2-Betti number of G. In this paper we address the questions of when Qapproximation and Fp-ap...