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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? (Note: the group $G/H$ is also assumed infinite dimensional and the local cross section is required continuous.)

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    $\begingroup$ Just to clarify: what conditions do you want on the cross-section? Continuity? $\endgroup$
    – Yemon Choi
    Sep 6, 2014 at 20:19
  • $\begingroup$ Yes, of course. I forgot to mention that. The cross section needs to be continuous. I will edit the question. Thanks. $\endgroup$ Sep 7, 2014 at 5:48

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