by Define the homotopy g ( γ, x ) = γ x + (1 − γ ) ψ $^{1}$( x ) for γ ∈ [0 , 1]. Then the homotopy formula gives We define R = g$_{#}$ ([0 , 1] × T ) and Q$_{1}$ = g$_{#}$ ([0 , 1] × ∂T ). Then we get Finally, we map ψ 1 $_{#}$( ∂T ) forward to the ( d − 1)-skeleton of simplicial complex K with φ = φ$_{k}$$_{+1}$ to get ψ 2 $_{#}$( ∂T ) = φ$_{#}$ ( ψ 1 $_{#}$( ∂T )). For this purpose, consider the homotopy h ( γ, x ) from ψ 1 $_{#}$( ∂T ) to ψ 2 $_{#}$( ∂T ), i.e., We define P is a d -current whose boundary ∂P is contained in the ( d − 1)-skeleton of K . Define Q$_{2}$ = h$_{#}$ ([0 , 1] × ψ 1 $_{#}$( ∂T )). Using the homotopy formula, we get 41 ∂P = ∂ $^{(}$ψ 1 $_{#}$( T ) − h$_{#}$ ([0 , 1] × ψ 1 $_{#}$( ∂T )) ) = ψ 1 $_{#}$( ∂T ) − ∂h$_{#}$ ([0 , 1] × ψ 1 $_{#}$( ∂T )) = ψ 2 $_{#}$( ∂T ) ⊂ ( d − 1)-skeleton of K. P = ψ 1 $_{#}$( T ) − h$_{#}$ ([0 , 1] × ψ 1 $_{#}$( ∂T )) . (2.19) h ( γ, x ) = γψ 1 $_{#}$( x ) + (1 − γ ) ψ 2 $_{#}$( x ) for γ ∈ [0 , 1] . T − ψ 1 $_{#}$( T ) = ∂R + Q$_{1}$. (2.18) T − ψ 1 $_{#}$( T ) = ∂g$_{#}$ ([0 , 1] × T ) + g$_{#}$ ([0 , 1] × ∂T ) . f$_{1}$$_{#}$ ( T ) − f$_{0}$$_{#}$ ( T ) = ∂g$_{#}$ ([0 , 1] × T ) + g$_{#}$ ([0 , 1] × ∂T ) .