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We investigate dynamical analogues of the $L^2$-Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that the $L^2$-Betti numbers exactly measure the failure of addition formula for dynamical invariants.
We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, w
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems
We correct the proofs of the main theorems in our paper Limit theorems for Betti numbers of random simplicial complexes.
We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multis
Given a torus $E = S^{1} times S^{1}$, let $E^{times}$ be the open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th singular Betti number $h^{i}(mathrm{Conf}^{n}(E^{times}))$ of the unordered configuration space of $