That is:

  
    
      
        
          
            T
          
          
            n
          
        
        =
        
          
            
              
                
                  S
                  
                    1
                  
                
                ×
                ⋯
                ×
                
                  S
                  
                    1
                  
                
              
              ⏟
            
          
          
            n
          
        
        .
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
        
        )
        .