Related papers: A pro-algebraic fundamental group for topological …

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 show how to functorially attach continuous $p$-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over $\mathbb{C}_p$. This generalizes results by…

Let $X(\Delta)$ be the real toric variety associated to a smooth fan $\Delta$. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of $X(\Delta)$, (ii) to give necessary and sufficient…

The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of…

Motivated by applications to duality theorems for $p$-adic pro-\'etale cohomology of rigid analytic spaces, we study the category of Topological Vector Spaces in the setting of condensed mathematics. We prove that it contains, as full…

We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential…

Let $k$ be a field, $K/k$ a field extension, $X$ a connected scheme proper over $k$, $x_K\in X_K(K)$ lying over $x\in X(k)$, $\mathcal{C}_X$ and $\mathcal{C}_{X_K}$ the Tannakian categories over $X$ and $X_K$ respectively,…

Let $X$ be a topological space. We denote by $\pi_0(X)$ the set of connected components of $X$ and by $\Pi_1(U)$ the fundamental groupoid. In this paper we prove that for good topological spaces the assignments $U\mapsto\pi_0(U)$ and…

We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a…

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as…

Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or…

Let $G$ be a connected reductive algebraic group. Let $\mathcal{E}\rightarrow \mathcal{B}$ be a principal $G\times G$-bundle and $X$ be a regular compactification of $G$. We describe the Grothendieck ring of the associated fibre bundle…

Let $X$ be a projective, connected and smooth scheme defined over an algebraically closed field $k$. In this paper we prove that a tower of finite torsors (i.e., under the action of finite $k$-group schemes) can be dominated by a single…

We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces…

We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the…

Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we…

Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to…

We introduce the category of finite \'etale covers of an arbitrary schematic finite space $X$ and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the \'etale fundamental group…

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $\pi_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give…

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves.