Let $S^1$ denote the circle and consider for every integer $k$ the map $f_k:S^1\rightarrow S^1: z\mapsto z^k.$
How can I compute the degree $d(f_k)$? It is easy to find $d(f_0)=0$ and $d(f_1)=1$. But what for the integers $k$?
EDIT: I have an idea. We can write $H_1(S^1)=
I assume that $c=c_1+\cdots+c_k$ (why is their difference a $1$-boundary?). We divide $d$ in the same way.
Then applying $f_{\#}(c+d)=f_{\#}(\sum_j c_j+\sum_j d_j)= k (c+d)$.