Topological groups, Lie groups Locally compact abelian groups and Pontryagin duality Example

Torus as a group

Builds on Circle group Torus Pontryagin dual

Example

Let $n \ge 1$ and let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the circle group. The $n$ -torus is the product group $\mathbb{T}^n$ with coordinatewise multiplication and the product topology. It is a locally compact abelian group. It is abelian because $\mathbb{T}$ is, Hausdorff as a subspace of $\C^n$ , and compact as a closed bounded subset of $\C^n$ . The map $(t_1, \dots, t_n) \mapsto (e^{2\pi i t_1}, \dots, e^{2\pi i t_n})$ is a continuous surjective homomorphism $\R^n \to \mathbb{T}^n$ with kernel $\Z^n$ , and it induces an isomorphism of topological groups $\R^n / \Z^n \cong \mathbb{T}^n$ .

Remarks

As a topological space, $\mathbb{T}^2$ is the torus surface; the group structure is the extra data. The torus $\mathbb{T}^n$ is a compact connected abelian Lie group of dimension $n$ , and every compact connected abelian Lie group is isomorphic to some $\mathbb{T}^n$ . Its Pontryagin dual is $\Z^n$ : the continuous characters are $(z_1, \dots, z_n) \mapsto z_1^{k_1} \cdots z_n^{k_n}$ with $(k_1, \dots, k_n) \in \Z^n$ . Fourier analysis on $\mathbb{T}^n$ is the theory of multiple Fourier series.

Reading path

To understand this page from the ground up, read these 70 pages in order. The foundations come first.

Binary operation definition
Element definition
Group concept
Pontryagin dual concept
Proposition concept
Real numbers object
Commutativity definition
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Abelian group definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Circle group object
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Product topology definition
Preimage of a set definition
Metric space definition
Codomain definition
Function composition definition
Injective function definition
Equivalence class definition
Open ball definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Continuous map of topological spaces definition
Open cover definition
Closed set definition
Complement of an open set is closed statement
Homeomorphism definition
Compact space definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Continuous image of a compact set is compact statement
Compact subsets of a Hausdorff space are closed statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Torus as a group example

Where this leads

1 page builds on this one.

Maximal torus

Tag	`XNAWD33HN1`
URI	`m2://XNAWD33HN1`
Kind	example
Writing	W0 stub
MSC	22B05, 22E15
Instance of	Locally compact abelian group
Uses	Circle group, Torus, Pontryagin dual
Changed	2 November 2026

Torus as a group

Example

Remarks

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