Related papers: A Comparison Theorem For The Pro-\'etale Fundament…
Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em…
The aim of the paper and of a wider project is to translate main notions of anabelian geometry into the language of model theory. Here we finish with giving the definition of the \'etale fundamental group $\pi^{et}_1(X,x)$ of a non-singular…
We construct the pro-Nisnevich topology, an analog of the pro-\'etale topology. We then show that the Nisnevich $\infty$-topos embeds into the pro-Nisnevich $\infty$-topos, and that the pro-Nisnevich $\infty$-topos is locally of homotopy…
Consider a connected topological space $X$ with a point $x \in X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite dimensional (flat) vector bundles on $X$ and its Tannakian dual $\pi_K (X,x)$…
We prove the following criterion for the pro-representability of the deformation cohomology of a commutative formal Lie group. Let f be a flat and separated morphism between noetherian schemes. Assume that the target of f is flat over the…
The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $\alpha(X)$ of compact families of open subsets of a…
We prove that there is an adjunction between what we call \'etale topological categories and restriction quantal frames that leads to an adjunction with a category of complete restriction monoids. This generalizes the adjunction between…
We introduce the notion of \emph{topo-symmetric extensions} of topological groups, a new generalization of classical group extensions that incorporates both topological and symmetry constraints. We define morphisms between such extensions,…
In this paper, we are interested in study subgroups of topologized fundamental groups and their influences on generalized covering maps. More precisely, we find some relationships between generalized covering subgroups and the other famous…
We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and…
Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…
We propose and develop a theory that allows to characterize epimorphisms of profinite groups in terms of indecomposable epimorphisms.
We describe the topology of a general polynomial mapping $f:\Bbb C^2\to\Bbb C^2.$
Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first…
We present a relative form of the Toponogov comparison theorem.
The purpose of this paper is to prove a basic $p$-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a $p$-adic field: $p$-adic pro-\'etale cohomology, in a stable range, can be…
We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. Then use this additional structure to analyze geometric/ topological properties of stacks.…
In this paper we analyse the topological group cohomology of finite-dimensional Lie groups. We introduce a technique for computing it (as abelian groups) for torus coefficients by the naturally associated long exact sequence. The upshot in…
Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically more suitable to study well-known problems in groups theory. We…
The recently introduced A-homotopy groups for graphs are investigated. The main concern of the present article is the construction of an infinite cell complex, the homotopy groups of which are isomorphic to the A-homotopy groups of the…