3-manifold
3-torus

The 3-dimensional torus is the product of 3 circles. That is:

The 3-torus, T can be described as a quotient of R under integral shifts in any coordinate. That is, the 3-torus is R modulo the action of the integer lattice Z (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.

A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
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