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
        
        )
        .
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 n-dimensional torus, often called the n-torus or hypertorus 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.