Contents

group theory

# Contents

## Idea

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

## Recognition

###### Proposition

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

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

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

include the following:

## Examples

###### Example

For $\mathcal{H}$ an infinite-dimensional separable complex Hilbert space, the coset space of the unitary group U(ℋ) by its circle subgroup U(1) is the projective unitary group $\mathrm{U}(\mathcal{H})/U(1) \; \simeq$ PU(ℋ), both in their weak/strong operator topology. This coset space coprojection falls outside the applicatibility of Prop. , and yet it does admit local sections (Simms 1970, Thm. 1).

## Counter examples

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