Show $\mathbb C^*\cong\mathbb R^*\times\mathbb T\cong\mathbb C/\mathbb Z$.

Question

I have to show that $\mathbb C^*\cong\mathbb R_{>0}\times\mathbb T\cong\mathbb C/\mathbb Z$. I've already shown that $\mathbb C^*\cong\mathbb R_{>0}\times\mathbb T$. So I either have to show that $\mathbb C^*\cong\mathbb C/\mathbb Z$ or $\mathbb R_{>0}\times\mathbb T\cong\mathbb C/\mathbb Z$. I'm opting for the first one.

One theorem I know, is that if $f\colon G_1\to G_2$ is a surjective homomorphism (where $G_1$ and $G_2$ are groups), then $G_1/\ker f\cong G_2$. Maybe if I could find a surjective homomorphism $g\colon\mathbb C\to\mathbb C^*$ with $\ker f=\mathbb Z$, then I'm done. But I can't think of any.

Or maybe I should find a direct isomorphism between $\mathbb C^*$ and $\mathbb C/\mathbb Z$. In any case, I'm clueless. Could someone me a hint?

Answer 1

Hint: consider $g:\mathbb{C}\rightarrow \mathbb{C}^*$ defined by $g(z)=\exp(2\pi i z)$, its kernel is $\mathbb{Z}$ and it is surjective.

Answer 2

Although essentially the same as Tesmo's answer, here is a way to give the other isomorphism. The isomorphism $\mathbb R_{>0}\times\mathbb T\cong\mathbb C/\mathbb Z$ is given as follows. The isomorphism $\mathbb C\cong\mathbb R\times\mathbb R:a+bi\mapsto (a,b)$ gives $\mathbb C/\mathbb Z\cong(\mathbb R\times\mathbb R)/(\mathbb Z\times0)\cong\mathbb R\times\mathbb R/\mathbb Z$.

Now, we finish off by observing the isomorphism $\exp\colon\mathbb R\to\mathbb R_{>0}$ and the sujection $\mathbb R\to\mathbb T:t\mapsto\cos(2\pi t)+i\sin(2\pi t)$ with kernel $\mathbb Z$ gives an isomorphism $\mathbb R/\mathbb Z\cong\mathbb T$. An advantage to this approach is that we do not use complex exponentiation explicitly; we only need real exponentiation and basic properties of trigonometric functions.