Group of homomorphisms with real target and circle target Denote by $\mathrm{Hom}$ continuous group homomorphisms. Fix the quotient homomorphism $\mathbb{R}\to S^1$. 
Can one characterize those topological (e.g., locally compact, and in particular discrete) groups $G$ such that the induced map $\mathrm{Hom}(G,\mathbb{R})\to\mathrm{Hom}(G,S^1)$ is bijective? 
Clearly this holds for $G$ if and only if it holds for the abelianization $G/\overline{[G,G]}$, so we can assume that $G$ is abelian and Hausdorff.
 A: If $G$ is a simply-connected Lie group, then any Lie group morphism $G\rightarrow S^1$ factors through the universal cover $\mathbb{R}\rightarrow S^1$ to give a Lie group morphism $G\rightarrow\mathbb{R}$ (by requiring the identity in $G$ to be sent to $0$). This gives a bijection between the Lie group $Hom$ sets. Is this the sort of answer you were seeking?
A: I guess you mean continuous homomorphisms. Note that the condition only depends on the abelianization $G/\overline{[G,G]}$; let $H$ be its Pontryagin dual. By Pontryagin duality it's equivalent to determine for which LCA groups $H$ every element lies in a 1-parameter subgroup. These groups $H$ have been characterized by Dixmier (*): those of the form $\mathbf{R}^k\times D$ where $D$ is a discrete abelian group such that $\mathrm{Ext}^1_\mathbf{Z}(D,\mathbf{Z})=0$. This fully characterize groups with the required property.
(*) J. Dixmier. Quelques propriétés des groupes abéliens localement compacts. Bull. Sci. Math. (2) 81 (1957) 38-48.
