We consider various notions of Mayer–Vietoris squares in algebraic geometry. use these to generalize a number gluing and push out results Moret-Bailly, Ferrand–Raynaud, Joyet Bhatt. An important intermediate step is Gabber's rigidity theorem for henselian pairs, which our methods give new proof of.

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips-Sela and Dunwoody-Sageev and ...

In this article we present a new algorithm for creating simplicial Vietoris-Rips complexes that is easily parallelizable using computation models like MapReduce and Apache Spark. The algorithm does not involve any computation in homology spaces.

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor on the category of sets. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. T...

We investigate the structure of the singular part of the second bounded cohomology group of amalgamated products of groups by constructing an analog of the initial segment of the Mayer-Vietoris exact cohomology sequence for the spaces of pseudocharacters.

We briefly review differential forms on manifolds. We prove homotopy invariance of cohomology, the Poincaré lemma and exactness of the Mayer–Vietoris sequence. We then compute the cohomology of some simple examples. Finally, we prove Poincaré duality for orientable manifolds.

We discuss gluing of objects and gluing of morphisms in tensor triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.

We refine and stratify the standard separation properties to produce a descending hierarchy between T3 and T1. The interpolated properties are related to the patch properties and the Vietoris modifications of the parent space.

The many different point-sensitive Vietoris modifications of a topological space have been around for many years. These constructions produce what are often called hyperspaces of a space. The point-free version, which is rather more modern, was introduced in [3] and reworked in [4]. Since then nothing much worth reading has been written about the construction, although [6] is an exception to th...