The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

We study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.

The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

We study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

On the universal cover and the fundamental group of an RCD*(K,N)-space

The main goal of the paper is to prove the existence of the universal cover for {\mathsf{RCD}^{*}(K,N)}-spaces. This generalizes earlier work of [43, 44] on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of {\mathsf{RCD}^{*}(K,N)}-spaces. These are the first topological results for {\mathsf{RCD}^{*}(K,N)}-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).

HOMOGENEOUS SPACE FIBRATIONS OVER SURFACES

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

On Simple Connectedness of Minimal Representation-Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Simple connectedness of one space of complex-valued functions

The spaces $\mathbb{C}{\Omega}_n$ have been defined. It has been established that the spaces $\mathbb{C}{\Omega}_n$, $n \geqslant 2$ are simply connected.

Relatively very free curves and rational simple connectedness

Rational connectedness is an algebro-geometric analogue of path connectedness depending crucially on the existence of special rational curves called

The Equilibrium Manifold

This chapter addresses the global structure of the equilibrium manifold E. It starts by motivating these global properties for their economic interest. These global properties can be of a topological nature like pathconnectedness, simple connectedness, and contractibility. They can also take a more practical form like the existence of global coordinate systems for the points of the equilibrium manifold in the same way the points at the surface of the earth can be located through their longitude and latitude. The chapter identifies an important subset of the equilibrium manifold, the set of no-trade equilibria. This enables us to uncover the remarkable structure of the equilibrium manifold as a collection of linear spaces parameterized by the no-trade equilibria. This remarkable structure is then applied to define a global coordinate system for the equilibrium manifold.