I am reading on magnetic monopoles from a variety of sources, eg. the Jeff Harvey lectures.. It talks about something called the winding $N$, which is used to calculate the magnetic flux. I searched the internet but am not being able to understand the calculation done in this particular case.
Then the author says that
Now $\Phi$ restricts to a map $\Phi : S_\infty^2 → S^2$ , where the target is the unit sphere in $su(2)$. This map has some degree $N$ , and it is easy to verify that the right-hand side of the above equation is $−2\pi$ times this. Therefore $g = −2\pi N$ .
What is $N$, the winding number also called as the degree on the map? By what i have learnt, it is the number of times you wind an object unto the another, then shouldn't the integral be $N*4\pi$, as $4\pi$ is the surface area of $S^2$.