Topological groups

A topological group is what one might expect from the name, a group $G$ equipped with a topology. We ask that the topology is nice with respect to the group structure. Namely, we require that the binary operation on the group $G \times G \to G$, and the inverse $G \to G$ are continuous.

One simple group is the circle group $\mathbb{T} = \{z \in \mathbb{C} : |z| < 1\}$ equipped with multiplication. To make this a topological group this group can be equipped with the standard topology coming from $\mathbb{C}$. Complex multiplication and inverses are clearly continuous on $\mathbb{T}$. As we will soon see, this group turns out play a central role in Pontryagin duality.

The next step is to define dual groups and state the Pontryagin duality theorem but first we define locally compact (abelian) groups.

Locally compact groups

A topological group $G$ is called locally compact if and only if there is a compact neighbourhood containing the identity. That is, if there exists some open set $U \ni e$ whose closure is compact in the topology of $G$.

For example, the circle group $\mathbb{T}$ is locally compact. Simply take $U = B_\epsilon(1) \cap \mathbb{T}$ for small $\epsilon > 0$ such that $\bar{U}$ is just some closed connected piece of the circle.

There are many more examples such as; $\mathbb{R}^n$ with addition and standard topology, and any finite abelian group equipped with the discrete topology.

Dual group

The dual of a group is analogous to the dual of a vector space but there are some subtle differences.

If $G$ is a locally compact abelian group then consider the set of continuous group homomorphisms $f : G \to \mathbb{T}$. Such functions are called characters of the group $G$ and collectively form a locally compact abelian group, called the dual group $G^\wedge$.

For a finite dimensional vector space $V$ over a field $\mathbb{K}$ we usually define the dual space as $Hom(V, \mathbb{K})$. Likewise, for a locally compact abelian group the dual group is $Hom(G, \mathbb{T})$.

For example, the integers with addition $(\mathbb{Z}, +)$ is isomorphic to the dual of the circle group $(\mathbb{T}, \cdot)$. Typically one proves this by showing that characters on $\mathbb{T}$ are of the form $z \mapsto z^n$ for some $n \in \mathbb{Z}$.

Pontryagin duality theorem

The dual of $G^\wedge$ is canonically isomorphic to $G$.

Canonical means that there is a natural isomorphism from $\phi: G \to (G^\wedge)^\wedge$. That is, $\phi(x) = \{\chi \mapsto \chi(x)\}$ for all $x \in G$, $\chi \in G^\wedge$.