If $G$ is a connected topological group, and $H$ is a closed subgroup of $G$, such that $G/H$ is simply connected, does it follow that $H$ is connected?
(In the particular class of cases I have in mind, $G$ is compact, and $G/H$ is homeomorphic to a convex set in $\mathbb{R}^n$. )
At least if $G, H$ are Lie groups, there is a long exact sequence
$$\dots \to \pi_1(G/H) \to \pi_0(H) \to \pi_0(G) \to \pi_0(G/H)$$
which shows that the answer is yes. In fact, the long exact sequence shows that if $G/H$ is contractible then the map $H \to G$ is a weak homotopy equivalence, and hence if $G, H$ are Lie groups it is a homotopy equivalence by Whitehead's theorem.