Let $q: S^n \rightarrow S^n $ be the map that quotients the lower hemisphere into the south pole $s$ . I am asked to calculate the degree of this map.
If $D_+ $ denotes the upper hemisphere then $q : D_+ \rightarrow S^n - s$ is a homeomorphism and hence by the local degree formula the degree is $1$ or $-1$.
I tried showing this map does not send any point to it's antipode and hence it is homotopic to identity. Thus it must have degree $1$.
This is intuitively quite clear to me but I don't know how to write up a formal proof. Any insight/corrections/help will be greatly appreciated.