Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of Rn gives a splitting, via the linear maps, as above):

  
    
      
        1
        →
        
          Homeo
          
            0
          
        
        ⁡
        (
        
          
            T
          
          
            n
          
        
        )
        →
        Homeo
        ⁡
        (
        
          
            T
          
          
            n
          
        
        )
        →
        
          MCG
          
            TOP
          
        
        ⁡
        (
        
          
            T
          
          
            n
          
        
        )
        →
        1.
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
        
        )
        .