Editing Winding number (section)

===Topology===
In [[topology]], the winding number is an alternate term for the [[degree of a continuous mapping]].  In [[physics]], winding numbers are frequently called [[topological quantum number]]s.  In both cases, the same concept applies.

The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is [[homotopy equivalent]] to the [[circle]], such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps <math>S^1 \to S^1 : s \mapsto s^n</math>, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of [[homotopy class]]es of maps from a circle to a [[topological space]] form a [[Group (mathematics)|group]], which is called the first [[homotopy group]] or [[fundamental group]] of that space. The fundamental group of the circle is the group of the [[integers]], '''Z'''; and the winding number of a complex curve is just its homotopy class.

Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes [[Pontryagin index]].