Homotopy classes of maps to Lie groups

Question

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.)

Answer 1

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 $U(n)\to U(n+1)\to S^{2n+1}$.

Answer 2

Topology of Lie Groups, I and II certainly has the homotopy groups up a ways.

Answer 3

Notice that since $G$ is a topological group, we have $$\pi_0[X,Y] = \pi_1 [X, BG]$$ Here $[X,Y]$ denotes the mapping space with standard compact-open topology, $BG$ is the classifying space for group $G$ (a connected space, such that its based loop space is homotopy equivalent to $G$). Since $G$ is a connected group, we have $\pi_0(BG) = \pi_1(BG) = 0$. For $k>1$ all $\pi_k(BG)$ are abelian and $\pi_k(BG) = \pi_{k-1}(G)$.

There exists a so-called Federer spectral sequence, which is a homological second quadrant spectral sequence, having $$E^2_{p,q} = H^{-p}(X,\pi_{q}(Y))$$ for $p+q\geqslant 0$ and $E^2_{p,q} = 0$ otherwise. Here (if $\pi_1(Y)$ acts trivially on higher homotopy groups of $Y$) $H^{-p}$ groups are singular cohomology of $X$ with coefficients in corresponding (abelian) homotopy groups of $Y$. Under suitable finiteness conditions this sequence converges to $\pi_{p+q}([X,Y],f)$, where $f\colon X\to Y$ is some specific mapping, considered as a base point in mapping space (the spectral sequence depends on path component). Sufficient convergence conditions include $X$ being a finite CW-complex. Since we are interested in the loop space $\Omega [X, BG]$, we should consider $f\colon X \to BG$ that is a trivial map. Also $\pi_1(BG) = 0$, so all conditions of the stated theorem are satisfied.

Obviously, if $\dim X = n$, then to calculate $\pi_i([X,Y])$ for $i\geqslant k$ we need to know only first $n+k$ homotopy groups of $Y$, so in principle you can find anything you want to know about the mapping space. There is also a standard problem that a spectral sequence generally gives not exactly the groups it converges to, but only some graded groups associated to the corresponding filtration.

H. Federer, A study of function spaces by spectral sequences, Trans. Amer. Math. Soc. , 82 (1956) pp. 340–361