In this paper, the assembly maps in algebraic Kand L-theory for the family of finite subgroups are proven to be split injections for word hyperbolic groups. This is done by analyzing the compactification of the Rips complex by the boundary of a word hyperbolic group.

The Vietoris powerlocale V X is a point-free analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale V X whose points are those points of V X that (considered as sublocales of X) satisfy a constructively strong connectedness property. V c is a strong monad on the category of locales. The strength gives rise to a product map × : V X × V Y → V (X × Y ), showing that ...

In [7] we introduced the category MKHaus of modal compact Hausdorff spaces, and showed these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in [7] we introduced the categories MKRFrm and MDV of modal compact regular frames, and modal de Vries algebras as algebraic c...

We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and approximate computation in comparison to raw point sets. This is achieved by examining the sublevel sets of the kernel distance, which isomorphically map to s...

The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach norm topology via weak* (when considering a predual $L^{1}$-functions). We prove that sequence $W_1,W_2,W_3,\ldots$ converges in distance if and only have equality sets accumulation points limit all sequences $W_1',W_2',W_3',\ldots$ are weakly isomorphic to $W_1,W_2,W_3,\ldots$. further give s...

In recent years, a new approach to data analysis has been developed, based on topological methods. The basic idea is to understand structures in a (large) set of data in a (high-dimensional) space, by associating to it a simplicial topological space and studying its topology. The starting point is a set of data with a proximity parameter (such as a distance function). The simplicial complex (Vi...

We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our notion of bisimulation is sound and complete for behavioural equivalence, and generalizes Vietoris bisimulations [3]. The main result of our paper is that bisimulation for a Stone coalgebra is the topological closure of bisimulation for the underlying Set coalgebra.

We prove that if a group G is systolic, i.e. if it acts properly and cocompactly on a systolic complex X, then an appropriate Rips complex constructed from X is a finite model for EG.