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William of Baskerville
William of Baskerville
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Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions? (TheNote: the group $G/H$ is also assumed infinite dimensional and the local cross section is required continuous.)

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions? (The group $G/H$ is also assumed infinite dimensional.)

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions? (Note: the group $G/H$ is also assumed infinite dimensional and the local cross section is required continuous.)

added 56 characters in body
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William of Baskerville
William of Baskerville
  • 985
  • 4
  • 13

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions? (The group $G/H$ is also assumed infinite dimensional.)

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions?

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions? (The group $G/H$ is also assumed infinite dimensional.)

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William of Baskerville
William of Baskerville
  • 985
  • 4
  • 13

Local cross sections in infinite dimensional groups

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples with $H$ totally disconnected. My question is the following. Assuming that $H$ is connected (in fact, the case when $H$ is the $\aleph_0$-dimensional torus is sufficient for me), does there exist a local cross section in this case? If not in general, are there any useful sufficient conditions?