nLab coset space coprojection admitting local sections




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

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


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topological homotopy theory

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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.



(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:



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 U()/U(1)\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 GG/H G \to G/H without local sections are given in Karube 58, Sec. 3.


Last revised on September 19, 2021 at 14:07:38. See the history of this page for a list of all contributions to it.