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I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup.

If $G$ is a locally compact Abelian group (second countable) and $H$ is a closed subgroup then $G$ and $H$ inherit a natural translation-invariant metric. You can use this to induce a translation-invariant metric on $G/H$.

Does there exist a neighbourhood $N$ of the identity in $G/H$ and a continuous map $f\colon G/H\to G$ such that $f(x)\in x$ for all $x\in N$?

EDITED to incorporate simplifying suggestion of Emil Jerabek and correction of Mark Schwarzmann.

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  • $\begingroup$ WLOG $X=G/H$, $f=\operatorname{id}$, and $x=0$. $\endgroup$
    – Emil Jeřábek
    Commented Apr 12, 2011 at 16:06
  • $\begingroup$ I agree that $X=G/H$ and $x=0$ wlog. Not sure what you mean by $f=id$ though. $f$ is supposed to map $G/H$ into $G$. $\endgroup$
    – Anthony Quas
    Commented Apr 12, 2011 at 16:42
  • $\begingroup$ It does now, but in the original version of the question which I was commenting on, $f$ was a function from $X$ to $G/H$, and what you now call $f$ was called $F$. $\endgroup$
    – Emil Jeřábek
    Commented Apr 12, 2011 at 17:14
  • $\begingroup$ What do you mean by $x \in f(x) + H$ for all $x \in N$? $x$ is a coset of $H$, what does it mean for it to belong to another coset of $H$? $\endgroup$
    – Mark
    Commented Apr 12, 2011 at 18:49

1 Answer 1

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You are essentially asking for a local cross section for the fiber bundle $(G,\pi: G \to G/H)$. According to this article1, it is known that such a section exists if the group $G$ is Lie, or more generally, LCSC and with finite covering dimension. I'm not an expert on the subject, but it seems to me that there exist metrizable LCSC abelian groups with infinite covering dimension, so this doesn't seem to give a complete answer to your question. Perhaps someone who knows more on the subject can illuminate this point, or provide a more recent reference.

1On the Universal Space for Group Actions with Compact Isotropyby Wolfgang L Lück and David Meintrup

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  • $\begingroup$ See also this related question, which specifies some other conditions under which a local section exists: mathoverflow.net/questions/57015/… $\endgroup$
    – Mark
    Commented Apr 12, 2011 at 22:43
  • $\begingroup$ Thanks for the info Mark. In the main example where I want it, the conclusion is trivial but I am interested to see how much it generalizes. $\endgroup$
    – Anthony Quas
    Commented Apr 14, 2011 at 15:54
  • $\begingroup$ Probably the meat is the case when $G$ is compact abelian. In this case its covering dimension is the $\mathbf{Q}$-rank of its Pontryagin dual (which is then abelian discrete). $\endgroup$
    – YCor
    Commented Jun 28, 2021 at 12:32

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