Relation between homotopy theory and symmetry transformation of the Lagrangian What is the relation between the symmetry transformations of the Lagrangian and homotopy theory? If yes, how? Not sure if this is a math or physics questions. References would be very helpful. 
 A: Maybe this is an answer on the question you've asked.
The main reason of applications of homotopy theory in QFT is the requirement of finiteness of an action. Suppose that we want to start from the field configuration for which an action is finite, and then to discuss perturbations near such configuration. The most simple way to satisfy the requirement of finiteness of action is to start from trivial configuration, but in principle we have to find all of such configurations, including nontrivial.
Suppose we have different (in some sense, see below) field configurations for which an action is finite. If we can't deform field configuration so that it is continuously transformed into the other one, than these two field configurations are topologically inequivalent; the role of barrier, which makes continuous transformation impossible, is played by potential barrier, for which an action becomes infinite. If we can, then we state that two such configurations are belonged to one homotopy class of equivalence. The set of all homotopy classes forms the homotopy group.
What is the connection between the symmetry of lagrangian and homotopy group? In general, the group symmetry defines the homotopy group, while the Maurer-Cartan invariant defines the set of all homotopy classes of such group.
As an example, suppose you have the Skyrme action with (pseudo)scalar fields $\pi^{a}$ in euclidean spacetime:
$$
\tag 1 S = \int d^{d}x\left( \frac{1}{2}\partial_{i}\pi^{a}\partial_{i}\pi_{a} - b(\partial_{i}\pi^{a}\partial_{j}\pi_{a})^{2}\right)
$$
Goldstone fields $\pi_{a}(\mathbf x)$ is the paramterization of the symmetry group $G/H$, where initial group $G$ is spontaneously broken to $H$. As follows from $(1)$, for making an action finite, the spatial derivatice of fields $\partial_{i}\pi_{a}$ have to decrease on spatial infinity faster than $|\mathbf x|^{-\frac{d}{2}}$; in general, this means that fields $\pi_{a}$ have to tend to the constant value. Since each of $\pi^{a}$ in a given point forms factor space $G/H$, then we always can change this constant value to the wished one. So that $\pi_{a}(\mathbf x )$ is nothing but mapping of $d$-dimensional space, in which the ($d-1$ dimensional) sphere $x \to \infty$ is the one point, on the manifold $G/H$ of all field values. Since such space is topologically equivalent to $d$-dimensional sphere $S_{d}$, so that fields configurations $\pi_{a}$ may be classified by topologically inequivalent mappings $S_{d} \to G/H$, which form the homotopy group. For example, for $G\sim SU_{L}(3)\times SU_{R}(3)$ and $H \sim SU_{V}(3)$ we have that $G/H \sim SU(3)$, and homotopy group $\pi_{3}(SU_{3}) = Z$ is nontrivial, so that there exist nontrivial topological configurations (skyrmions) for which an action is finite. In realistic chiral perturbation theory they may be associated with baryons and resonances. The Maurer-Cartan invariant which defines the homotopy class is
$$
n = \int d^{3}x\epsilon^{ijk}\text{Tr}\left(U\partial_{i}U^{-1}U\partial_{j}U^{-1}U\partial_{k}U^{-1} \right)
$$
(here $U \equiv e^{i\frac{\pi_{a}t_{a}}{f_{\pi}}}$ is such nontrivial configuration), and it accidentally coincides with the baryon number.
