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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 $H_1(S^1)=<c+d>$ where $c:I\rightarrow S^1:t \mapsto e^{\pi it}$ and $d:I\rightarrow S^1:t \mapsto e^{\pi it+\pi i}$. Divide $c$ in $k$ equal parts $c_j:I\rightarrow S^1:t \mapsto e^{\frac{\pi j t}{k}+\frac{\pi j}{k}}$ for $j=1,\cdots, k$.

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 $f_{\#}(c+d)=f_{\#}(\sum_j c_j+\sum_j d_j)= k (c+d)$.

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The $k$ is the degree of $f$. What is the degree of a map $f:S^1\rightarrow S^1$? Intuitively, it is the number of times $f$ wraps $S^1$ around itself. Formally, $f$ induces a map $f_*:\mathbb{Z}\rightarrow\mathbb{Z}$. What is $f_*(1)$?

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I have edited the question – Nadori Jun 12 '12 at 0:39

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