Let $c$ be a cycle in $\mathbb C^{\times}$ and $c_n:[0,1]\to \mathbb C^{\times},c_n(t)=e^{2\pi int}$. Show that $c$ and $c_n$ are homologous.
They are homologous if the winding number $win(c-c_n,z)=0,\forall z\in\mathbb C\setminus\mathbb C^{\times}$. It follows that $z=0$. $$win(c-c_n,z=0)=win(c,0)-win(c_n,0)$$$$\frac{1}{2\pi i}\left(\int_c\frac{dw}{w-z}-\int_{c_n}\frac{dw}{w-z}\right)$$$$\frac{1}{2\pi i}\left(\int_c\frac{dw}{w}-\int_{c_n}\frac{dw}{w}\right)=\frac{1}{2\pi i}\left(\int_c\frac{dw}{w}-2\pi in\right)$$ I think there is a mistake because it is not obvious that $\int_c\frac{dw}{w}=2\pi in$.