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.)



Homotopy classes of maps to Lie groups
If the dimension of $M$ is low relative to that of $G$ then the calculation of $[M,G]$ typically reduces to stable homotopy theory or generalised cohomology, for which many methods are known. For example, if $\dim(M)<2n$ then 
$$[M,U(n)]\simeq [M,U(\infty)] \simeq [\Sigma M,BU(\infty)] \simeq K(\Sigma M) $$
(where $K$ denotes complex $K$-theory). This can often be understood using explicit constructions with vector bundles, or using the Atiyah-Hirzebruch spectral sequence $H^\*(M;K^\*)\Longrightarrow K^\*(M)$. Similar methods work for $[M,O(n)]$, $[M,SU(n)]$, $[M,Sp(n)]$ and so on, provided that $n$ is large enough. If you want to consider small $n$ then it may be possible to work back from large $n$ using fibrations like $U(n)\to U(n+1)\to S^{2n+1}$. 


