I am a Ph.D. student of physics and now I have some problems regarding the evaluation of homotopy group of a specific structure.
In a paper, a specific topological structure is defined. The structure is a map from a circle to $GL(m,\mathbb{C})$.
The paper claims the first homotopy classes can be indexed by an integer which is given by:
$$\nu_1=\frac{1}{2\pi}\int_{-\pi}^{\pi}\text{Tr}[U(k)^{-1}i\partial_kU(k)]$$
where $U$ is the map which maps $k\in[-\pi,\pi]$ to invertible $m\times m$ complex matrices. $k=-\pi$ and $k=\pi$ are identified to form a circle.
I have some background in undergraduate topology and I know what a homotopy group is. However, in undergraduate class there's no metric hence no differential operator in topological spaces so I don't know where this formula comes from.
Can someone give me a reference or explanation of this formula?