TITLE: Smooth map homotopic to Lie group homomorphism
QUESTION [11 upvotes]: 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.

REPLY [8 votes]: As Igor shows, every endomorphism of a simple Lie group $G$ has degree $\in\{0,\pm 1\}$.
On the other hand, every compact Lie group admits self maps of other degrees.  Namely, the $k$-th power map $g\mapsto g^k$ has degree $k^r$, where $r$ is the rank of the group.  So, each $k$ with $|k|\geq 2$ gives an example of a smooth map which is not homotopy equivalent to a homomorphism.
One way to compute the degree of the $k$-th power map is as follows.  First, we can find an element $g\in G$ which lies in a unique maximal torus $T^r$ and which is also a regular value of the $k$-th power map.  The uniqueness of the maximal torus implies that all $k$-th roots of $g$ lie in $T^r$, so this reduces the degree calculation to $T^r$, where it is obvious.