nLab coset space coprojection admitting local sections

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:

• $G$ is any topological group

  and $H$ is a compact Lie group

  (Gleason 50, Thm. 4.1)

• $G$ is a locally compact topological group

  which is moreover a separable metric space of finite dimension

  and $H \subset G$ is a closed subgroup.

  (Mostert 53, Theorem 3, see also Karube 58, Theorem 2)

• $G$ is a (finite-dimensional) Lie group

  and $H$ is a closed subgroup

  (e.g. tom Dieck & Bröcker 1985, Thm. 4.3)

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.

Related concepts

• coset space

• principal bundle

• G-structure

• equivariant bundle

References

• Hans Samelson, Beiträge zur Topologie der Gruppenmannigfaltigkeiten, Ann. of Math. 2, 42, (1941), 1091 - 1137. (jstor:1970463, doi:10.2307/1970463)

• Andrew Gleason, Spaces With a Compact Lie Group of Transformations, Proceedings of the American Mathematical Society Vol. 1, No. 1 (Feb., 1950), pp. 35-43 (jstor:2032430, doi:10.2307/2032430)

• Paul Mostert, Local Cross Sections in Locally Compact Groups, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 (jstor:2032540, doi:10.2307/2032540)

• Takashi Karube, On the local cross-sections in locally compact groups, J. Math. Soc. Japan 10(4): 343-347 (October, 1958) (doi:10.2969/jmsj/01040343)

• Tammo tom Dieck, Theodor Bröcker, Thm. 4.3 on p. 33 of: Representations of compact Lie groups, Springer 1985 (doi:10.1007/978-3-662-12918-0)