For $n\in \mathbb Z_+$, let $T^{2n+1}$ be the torus and $U(N)$ be the unitary group, where $N$ is sufficiently large.
A physics paper on topological insulators claims the following:
For a map $g: T^{2n+1} \to U(N)$, we can associate an integer-valued winding number defined as $$ \frac{(-1)^n n!}{(2n+1)!} \left(\frac{i}{2\pi}\right)^{n+1} \int_{T^{2n+1}} \mathrm{Tr}\left( (g^{-1} dg)^{2n+1}\right) \in \mathbb Z.$$
Why is it integer-valued?
How can we interpret the above quantity as the winding number? (The dimensions of $T^{2n+1}$ and $U(N)$ are different, but what I know from the de Rham cohomology theory is that the degree of map is defined for two connected and compact manifolds with the same dimension.)
Is the above invariant related to the fact that $\pi_{2n+1}(U(N)) \simeq \mathbb Z$? (The physics paper also mentions this fact, but not explicitly states how the above invariant is related to $\pi_{2n+1}(U(N))$.)
Is the above formula valid if we replace the domain of $g$ to $S^{2n+1}$?