# Topological Classification of Band Insulators (in terms of Green's functions)

I am currently reading Topological Classification and Stability of Fermi Surfaces by Y. X. Zhao and Z. D. Wang (PRL 110, 240404 (2013)).

They remark that the Green's function (along the complex frequency axis) can be viewed as a mapping $S^p \to \mathrm{GL}(N,\mathbb{C})$, where $p$ is the co-dimension of the Fermi surface. It is then natural to classify the Fermi surface by the topological character of this mapping which falls into some instance of $\pi_p(\mathrm{GL}(N,\mathbb{C})$). In particular, one can define the winding number

$$N_p = C_p \int_{S^p} \mathrm{tr}~(G\textbf{d} G^{-1})^p,$$

where $C_p = -~p!~/~( (2p+1)! (2\pi i )^{p+1})$ .

I am now wondering how these ideas are related to the topology of band insulators:

1. Since the Brillouin zone is periodic, an insulating band represents a compact manifold of some co-dimension $p$ (similar to a Fermi surface). And hence I should be able to write down the winding number $N_p$ in terms of Green's functions. Is this a straightforward generalisation of the formula presented above?
2. I suppose that in this case the Green's function needs to be viewed along the real frequency axis in order to pick up the correct singularities. Is this correct?