Understanding quotient groups

Question

Admittedly this will be probably be a naive question, but here it goes:

Is it possible to flesh out in simple terms, for someone with little background in group theory, what it means to take the quotient group of $\mathbb{R}$ or $\mathbb{C}$ by a lattice $\Lambda$ or by $\mathbb Z$? How to evaluate this quotient? As in what comes out?

More precise examples:

• If $\Lambda$ is a lattice spanned by linearly independent primitive vectors $\mathbf{a}_1$ and $\mathbf{a}_2,$ then how to interpret the quotient: $\mathbb C /\Lambda?$

• Another example: the circle $S^1$ can be identified with the quotient $\mathbb R/2\pi\mathbb Z,$ but I fail to make this identification myself. Any help is much appreciated here.

Answer 1

You can think of a quotient as glueing things together. If you quotient out by $\Lambda$ then you consider every point that differs by an element of $\Lambda$ to be the same, i.e. glued together.

A simple example is modular arithmetic: $[0]_5=\{0, \pm 5, \pm 10, \pm 15\,\ldots\}$ are all the same, modulo five. They form the same residue class $[0]_5$. In terms of a quotient, this residue class is $\mathbb Z / 5\mathbb Z$.

In the case of $\mathbb C/\Lambda$ consider all points in $\mathbb C$, that differ only by an element of $\Lambda$, to be the same/glued. I assume your lattice is given by integer multiples of the complex numbers $u$ and $v$. That means $\Lambda = \{\lambda u + \mu v : \lambda,\mu \in \mathbb Z \}$.

Let the elements of $\mathbb C$ that differ from $z \in \mathbb C$ by addition of an element of $\Lambda$ by $[z]$. We have $$[z] = \{ z+ \lambda u + \mu v : \lambda,\mu \in \mathbb Z \}$$ In modular notation, $w \in [z]$ if, and only if, $w \equiv z \bmod \Lambda$.

The set $[z]$ forms a single element of the quotient $\mathbb C/\Lambda$. You'll After a while, you can convince yourself that $\mathbb C/\Lambda = \{ [\lambda u + \mu v] : 0 \le \lambda , \mu < 1 \}$.

Answer 2

The quotient $\mathbb{C}/\Lambda$ is a complex torus (see the picture for it on page $4$ here); and the group is isomorphic to the group $E(\mathbb{C})$ of an elliptic curve $E$. The isomorphism is given by $(x,y)\mapsto (\wp(z),\wp'(z))$, for the Weierstrass $\wp$-function, with the elliptic curve equation $$ \wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3. $$ Hence quotients $\mathbb{C}/\Lambda$ are tori, and tori are elliptic curves over $\mathbb{C}$.

Answer 3

It may help you to consider some examples of quotient groups that you are already comfortable in working with:

$\mathbb Z/n\mathbb Z$

The set of integers $\mathbb Z$ forms a group under addition. It contains a subgroup $n\mathbb Z=\{nk:k\in\mathbb Z\}$ consisting of the elements of $\mathbb Z$ that are divisible by $n$.

The quotient group is the set of cosets $$\{n\mathbb Z, 1+n\mathbb Z, 2+n\mathbb Z,\ldots,(n-1)+n\mathbb Z\}$$ where $m+n\mathbb Z =\{m+nk:k\in\mathbb Z\}$, which is a group under the operation inherited from $\mathbb Z$: $$(a+n\mathbb Z)+(b+n\mathbb Z) = (a+b)+n\mathbb Z$$ where you should notice that if $a+b\in n\mathbb Z$, then $(a+b)+n\mathbb Z = n\mathbb Z$.

This group is exactly the group of integers modulo $n$: write $$a\equiv b\pmod n$$ if and only if $n\mid(b-a)$. Then the map sending $a+n\mathbb Z$ to $a\pmod n$ is an isomorphism.

$\mathbb R/\mathbb Z$

Let $[\cdot]:\mathbb R\to [0,1)$ be the map which sends a real number $x$ to its decimal part (e.g. $231.35642\mapsto 0.34642$).

If $x,y\in \mathbb R$, then it is easy to check that $$[x]+[y]=[x+y]$$ so that in particular, $[\cdot]$ defines a group structure on $[0,1)$.

You should check that with this group structure, $[0,1)\cong \mathbb R/\mathbb Z$. $\mathbb R/\mathbb Z$ is therefore the group of real numbers where we consider two numbers as the same if they differ by an integer.

An important observation is that there is an isomorphism between $[0,1)$ and the unit circle in $S^1\subset\mathbb C^\times$ (both are groups under multiplication) given by $$[0,1)\to S^1 \\ x\mapsto e^{2\pi ix}$$

So $\mathbb R/\mathbb Z\cong S^1$, and can be viewed as a unit line in $\mathbb R$, where the two ends are glued together.

The quotient $\mathbb C/(\mathbb Z+\mathbb Zi)$ is a two-dimensional analgoue of this: it is a unit square in $\mathbb R^2$ with opposite sides glued together - or a torus $S^1\times S^1$

and $\mathbb C/\mathbb Z$ can be defined similarly for a general lattice.