winding number in several complex variables

Question

Is there any analogue of the concept of winding numbers in the theory of several complex variables? If so, can anyone provide me references for studying it?

Answer 1

One can view the winding number as actually an element of the infinite cyclic group $\Bbb Z$. Each number corresponds to an equivalence class of paths in $\Bbb C\setminus\{0\}$ (so basically paths around the origin) modulo homotopy (which begin and end at the same point, i.e. form a loop). Hence the winding number is generalized by the notion of a fundamental group for a (in general, pointed) topological space. The fundamental group $\pi_1(X,x)$ of a topological space around a point $x$ essentially encodes loops around the point $x$ whose group operation is concatenation of paths back-to-back. This basically encodes the "ways of going around" inside the space, where two ways are equivalent up to "wiggling."

The theory of covering spaces places fundamental groups in the context of monodromy, which is a way of describing the symmetry of the collection of ways to cover a space. Essentially this says that the paths in $\pi_1$ act as the automorphisms (deck transformations) of a universal covering map. Check the links for more information, this idea of monodromy actions is quite a beautiful idea.

The above information applies to topological spaces generally. In particular, the theory of several complex variables is positioned within Riemann surfaces and complex manifolds, which are particular kinds of topological spaces.