nLab
coset space coprojection admitting local sections

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Group Theory

Contents

Idea

Given a topological group GG and a subgroup HGH \subset G it is often oft interest to known that the coset space coprojection GG/HG \to G/H admits local sections. For instance, these yield canonical examples of HH-principal bundles, of H-structures and of equivariant bundles.

Recognition

Proposition

(sufficient conditions for coset space coprojections having local sections)
Let GG be a topological group and HGH \subset G a subgroup.

Then sufficient conditions for the coset space coprojection GqG/HG \overset{q}{\to} G/H to admit local sections, in that there is an open cover iIU iG/H\underset{i \in I}{\sqcup}U_i \to G/H and a continuous section σ 𝒰\sigma_{\mathcal{U}} of the pullback of qq to the cover,

G |𝒰 G σ (pb) iIU i = iIU i G/H, \array{ && G_{\vert \mathcal{U}} &\longrightarrow& G \\ & {}^{\mathllap{ \exists \sigma }} \nearrow & \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \mathllap{ \exists \; } \underset{i \in I}{\sqcup} U_i &=& \underset{i \in I}{\sqcup} U_i &\longrightarrow& G/H \mathrlap{\,,} }

include the following:

Counter examples

Examples of quotient coprojections GG/H G \to G/H without local sections are given in Karube 58, Sec. 3.

References

Last revised on September 6, 2021 at 09:04:48. See the history of this page for a list of all contributions to it.