Quotient Maps with Connected Fibers and the Fundamental Group
نویسنده
چکیده
In classical covering space theory, a covering map induces an injection of fundamental groups. This note reveals a dual property for quotient maps having connected fibers, with applications to orbit spaces of smooth vector fields and leaf spaces in general.
منابع مشابه
Jack S. Calcut Research Statement 1.2 Quotient Maps and the Fundamental Group 1.3 the Pullback Functor on Coverings
The concept of a topology for the fundamental group appears to have originated with Hurewicz (1935), but it has been actively studied only for the last ten years. The classical fundamental group of a pointed topological space X naturally inherits the quotient topology from the space of based loops in X endowed with the compact-open topology. The resulting space, called the topological fundament...
متن کاملA connectedness property of algebraic moment maps
Let K be a connected compact Lie group and M a Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k := (LieK). A theorem of Kirwan (implicitly in [Ki]) asserts: if M is connected and compact then the level sets of μ are connected. The purpose of this note is to prove such a statement in the category of algebraic varieties. First, we reformulate Kirwan’s the...
متن کاملThe Fundamental Group of Symplectic Manifolds with Hamiltonian Lie Group Actions
Let (M,ω) be a connected, compact symplectic manifold equipped with a Hamiltonian G action, where G is a connected compact Lie group. Let φ be the moment map. In [12], we proved the following result for G = S action: as fundamental groups of topological spaces, π1(M) = π1(Mred), where Mred is the symplectic quotient at any value of the moment map φ, and = denotes “isomorphic to”. In this paper,...
متن کامل$L$-Topological Spaces
By substituting the usual notion of open sets in a topological space $X$ with a suitable collection of maps from $X$ to a frame $L$, we introduce the notion of L-topological spaces. Then, we proceed to study the classical notions and properties of usual topological spaces to the newly defined mathematical notion. Our emphasis would be concentrated on the well understood classical connectedness...
متن کاملThe Fundamental Group of G-manifolds
Let M be a connected smooth G-manifold, where G is a connected compact Lie group. In this paper, we first study the relation between π1 (M) and π1 (M/G). Then we particularly focus on the case when M is a connected Hamiltonian G-manifold with an equivariant moment map φ. In [13], for compact M , we proved that π1 (M) = π1 (M/G) = π1 (Ma) for all a ∈ image(φ), where Ma is the symplectic quotient...
متن کامل