On fundamental groups of quotient spaces

Quotient Spaces modulo Algebraic Groups

In algebraic geometry one often encounters the problem of taking the quotient of a scheme by a group. Despite the frequent occurrence of such problems, there are few general results about the existence of such quotients. These questions come up again and again in the theory of moduli spaces. When we want to classify some type of algebraic objects, say varieties or vector bundles, the classifica...

Norms on Quotient Spaces

Fundamental Groups of One-dimensional Spaces

Let X be a metrizable one-dimensional continuum. In the present paper we describe the fundamental group of X as a subgroup of its Čech homotopy group. In particular, the elements of the Čech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental g...

On the Fundamental Groups of Products of Topological Spaces

The following proposition is true (1) Let G, H be non empty groupoids and x be an element of ∏ 〈G,H〉. Then there exists an element g of G and there exists an element h of H such that x = 〈g, h〉. Let G1, G2, H1, H2 be non empty groupoids, let f be a map from G1 into H1, and let g be a map from G2 into H2. The functor Gr2Iso(f, g) yields a map from ∏ 〈G1, G2〉 into ∏ 〈H1,H2〉 and is defined by the ...

On the Fundamental Groups of One–dimensional Spaces

We study the fundamental group of one–dimensional spaces. Among the results we prove are that the fundamental group of a second countable, connected, locally path connected, one–dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and that the fundamental group of a compact, one–dimensional, connected metric space embeds in an inverse li...