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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 $x\in f(x)+H$ f(x)\in x$ for all $x\in N$?

EDITED to incorporate simplifying suggestion of Emil Jerabek and correction of Mark Schwarzmann.
show/hide this revision's text 2 simplification based on comments; edited body

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 $G/H$.

Given a continuous function $f\colon X\to G/H\ $ and a point $x\in X$, does

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

EDITED to incorporate simplifying suggestion of Emil Jerabek.
show/hide this revision's text 1

Picking a representative in a continuous way

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 $G/H$.

Given a continuous function $f\colon X\to G/H\ $ and a point $x\in X$, does there exist a neighbourhood $N$ of $x$ and a continuous function $F\colon X\to G$ such that $f(x)=F(x)+H\ $ for $x\in N$?