Direct sum of topological groups

In mathematics, a topological group G is called the topological direct sum of two subgroups H₁ and H₂ if the map

${\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}$

is a topological isomorphism.

More generally, G is called the direct sum of a finite set of subgroups $H_{i},i=1,\ldots ,n$ of the map

${\begin{aligned}\prod _{i=1}^{n}H_{i}&\longrightarrow G\\(h_{i})_{i\in I}&\longmapsto h_{1}h_{2}\cdots h_{n}\end{aligned}}$

Note that if a topological group G is the topological direct sum of the family of subgroups $H_{i}$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family $H_{i}$.

Topological direct summands

Given a topological group G, we say that a subgroup H is a topological direct summand of G (or that splits topologically from G) if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K.

A the subgroup H is a topological direct summand if and only if the extension of topological groups

$0\to H{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}G/H\to 0$

splits, where $i$ is the natural inclusion and $\pi$ is the natural projection.

Examples

• Suppose that $G$ is a locally compact abelian group that contains the unit circle $\mathbb {T}$ as a subgroup. Then $\mathbb {T}$ is a topological direct summand of G. The same assertion is true for the real numbers $\mathbb {R}$