homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian). Two-sheeted cover The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of the torus is determined by the cross-ratio of the four points. n-dimensional torus The torus has a generalization to higher dimensions, the , often called the or for short. (This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n.) Recalling that the torus is the product space of two circles, the n-dimensional torus is the product of n circles. That is: The standard 1-torus is just the circle: . The torus discussed above is the standard 2-torus, . And similar to the 2-torus, the n-torus, can be described as a quotient of under integral shifts in any coordinate. That is, the n-torus is modulo the action of the integer lattice (with the action being taken as vector addition). Equivalently, the n-torus is obtained from the n-dimensional hypercube by gluing the opposite faces together. An n-torus in this sense is an example of an n-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. Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group G one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G. Toroidal groups are examples of protori, which (like tori) are compact connected abelian groups, which are not required to be manifolds. Automorphisms of T are easily constructed from automorphisms of the lattice , which are classified by invertible integral matrices of size n with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on in the usual way, one