I understand the definition and some properties, but i was just wondering, is there some intuition behind the degree of a map $f:S^n \rightarrow S^n$?
The identity has degree $1$, the antipodal map has degree $-1$. What are maps with degree $0$? Are they constant?
Best Answer
It helps to use the case $n = 1$ as a motivating example. The map $S^1 \to S^1$ in which the domain gets wrapped around the target $k$ times has degree $\pm k$, where the sign depends on whether or not you throw in a reflection. The same idea is more or less true for higher-dimensional spheres, though it gets slightly harder to visualise.
And yes, the special case of degree $0$ is the case in which you wrap around the target $0$ times. Up to a continuous deformation that's the same as a constant map.