Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$?

Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates to $\mathbb{C} / \mathbb{Z}$.

By "nice" I mean something like:

  • $\mathbb{R}/\mathbb{Z}$ is isomorphic to the circle group via the exponential map $\theta \mapsto e^{i\theta}$, and
  • $\mathbb{C}/\Lambda$ is a complex torus for $\Lambda$ an integer lattice (an integer lattice is a discrete subgroup of the form $\alpha\mathbb{Z} + \beta\mathbb{Z}$ where $\alpha,\beta$ are linearly independent over $\mathbb{R}$.)

Intuitively, it seems like it should be something like a circle or elliptic curve.

  • 1
    $\begingroup$ As a group, and even as a Lie group, it's $\mathbb{R} \times \mathbb{R}/\mathbb{Z} \cong \mathbb{R} \times S^1$. $\endgroup$ – Qiaochu Yuan May 23 '15 at 2:27

$\mathbb C/\mathbb Z$ is isomorphic to $\mathbb C/ n\mathbb Z$ for any non-zero complex number $n$.

You can show that $\mathbb C/\mathbb Z\cong (\mathbb C\setminus\{0\},\times)$ via the homomorphism $z \to e^{2\pi iz}$.

You can also think of this as $S^1\times\mathbb R$, which is topologically a cylinder.


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