Let $G$ and $H$ be connected Lie groups. A Lie group homomorphism $\rho:G\to H$ is a smooth map of manifolds which is also a group homomorphism.
Question: Can we find a smooth (or real-analytic) map $f:G\to H$ which is not homotopic to any Lie group homomorphism?
For example, if $G=H=S^1$, it seems the answer is no. For simplicity, we may begin with the same question but assuming some extra conditions, such like (i) $G,H$ are torus, (ii) $G,H$ are compact, etc.