Let $S^n\subset \mathbb{R}^{n+1}$ be the unit sphere, and $T^n=(S^1)^n$ the $n$-torus. Loday proved in 1 that every algebraic map $T^n\to S^n$ is null-homotopic. In particular, since $SU(2)\simeq S^3$, every algebraic map $T^3\to SU(2)$ is null-homotopic.
What can be said more generally about algebraic maps from $T^n$ to $SU(k)$, maybe assuming $k$ is large enough?
Has this been investigated in any way? Why is K-theory involved in Loday's proof and should I look into it if I want to have a go at this question?
[1] Jean-Louis Loday. Applications algébriques du tore dans la sphère et de Sp×Sq dans Sp+q.