{\displaystyle \mathbf {T} ^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}.} The 1-torus is just the circle: T1 = S1. The torus discussed above is the 2-torus, T2. And similar to the 2-torus, the n-torus, Tn can be described as a quotient of Rn under integral shifts in any coordinate. That is, the n-torus is Rn modulo the action of the integer lattice Zn (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.
At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra:

  
    
      
        
          MCG
          
            Ho
          
        
        ⁡
        (
        
          
            T
          
          
            n
          
        
        )
        =
        Aut
        ⁡
        (
        
          π
          
            1
          
        
        (
        X
        )
        )
        =
        Aut
        ⁡
        (
        
          
            Z
          
          
            n
          
        
        )
        =
        GL
        ⁡
        (
        n
        ,
        
          Z
        
        )
        .