nTorus

The n-torus, often written as T^n, is a mathematical object formed by the Cartesian product of n circles. It can be realized as the quotient R^n / Z^n, with Z^n acting by translations. This construction makes T^n a compact, connected smooth manifold of dimension n and a commutative Lie group under coordinate-wise addition modulo 1.

As a topological space, T^n has fundamental group Z^n and universal cover R^n. A standard flat metric on R^n descends to a flat Riemannian metric on T^n. Geodesics correspond to straight lines in R^n projected to the quotient. The space is orientable and serves as a basic example of a compact Lie group (S^1)^n.

In homology and cohomology, T^n has well-known structures: the k-th homology group is H_k(T^n) ≅ Z^{C(n,k)}, where C(n,k) is a binomial coefficient. The cohomology ring H^*(T^n; Z) is the exterior algebra Λ^*(Z^n), generated by n degree-1 classes. These algebraic features reflect the product structure of the torus.

Common special cases include T^1, which is the circle S^1, and T^2, the standard two-dimensional torus obtained by identifying opposite edges of a square. Higher n-tori, such as T^3, arise in various areas of mathematics and theoretical physics.

Applications of the n-torus appear in topology, dynamical systems, and physics. It serves as a fundamental example in the study of compact manifolds, torus actions in geometry, and as a simple model for phase spaces in integrable systems and certain string theory compactifications.