In Physics one often encounters maps from a certain manifold $M$ to a Lie group $G$. For example, in gauge theories, this gives a gauge transformation, wich is a symmetry of a theory. It is then important to give a homotopy classification of such maps. One of the physical motivations is that maps non-homotopic to constant identity give examples of "large gauge transformations", which can turn out to be not exactly symmetries of the theory, and the requirement for them to be a symmerty leads to certain quantisation conditions, for example the quantization of level $k$ in non-abelian Chern-Simons theory.

Typical manifolds are $\mathbb{R}^n$, which is trivial, $S^n$, which reduces to $\pi_n(G)$.

The general problem of classification of homotopy classes $[M,N]$ is surely very hard, but I thought there may be some results when $N=G$ is a Lie group (for me it seems that this fact brings a group structure to the set of classes via pointwise multiplication of representatives). Probably I am not really good at looking for references, so I decided to post a question here:

Does anybody know a reference where the set $[M,G]$ is discussed for $G$ a Lie group?

(I know that for $U(1)$ it is $[M,K(\mathbb{Z},1)]=H^1(M,\mathbb{Z})$, pretty usefull in physics, but thats all.)