Direct sum of topological groups

In mathematics, a topological group $G$ is called the topological direct sum of two subgroups $H_1$ and $H_2$ if the map
$\begin H_1\times H_2 &\longrightarrow G \\ (h_1,h_2) &\longmapsto h_1 h_2 \end$
is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition

More generally, $G$ is called the direct sum of a finite set of subgroups $H_1, \ldots, H_n$ of the map
$\begin \prod^n_ H_i &\longrightarrow G \\ (h_i)_ &\longmapsto h_1 h_2 \cdots h_n \end$
is a topological isomorphism.

If a topological group $G$ is the topological direct sum of the family of subgroups $H_1, \ldots, H_n$ 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 $K \leq G$ 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
$0 \to H\stackrel G\stackrel G/H\to 0$
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 $\mathbb$ as a subgroup. Then $\mathbb$ is a topological direct summand of $G.$ The same assertion is true for the real numbers $\R$Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. MR0637201 (83h:22010).

See also

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References

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Topological groups
Topology