I have two different Lie groups, one is an arcwise connected Lie subgroup
$G_1 \subseteq \mathbb{R}^m \times (S^1)^n$
and the other is
$G_2 = \mathbb{R}^k \times (S^1)^l$
While $G_1$, $m$ and $n$ are fixed, I can change $G_2$ by varying $k$ and $l$ as I please. I'm interested in continuous
$f : G_1 \to G_2$
$g : G_2 \to G_1$
and was wondering if anything at all can be said about $f$ or $g$? Ideally, I would like to find any condition for $l$ and $k$, so that $f$ or $g$ become "almost" homeomorphic or that both are surjective. By "almost" I'm trying to express that there exists an $f$ and $g$ for which
$(g \circ f)(p) = p$
is true except for a tiny finite subset. Is that possible?