Equation (2.19) gives ψ 1 $_{#}$( T ) = P + Q 2. Defining Q = Q$_{1}$ + Q$_{2}$ , Equation (2.18) gives Finally, we apply the bounds on the retraction described in Lemma 2.5.9 and the paragraph following this Lemma to the masses of the pushforwards. Noticing that diameter( σ ) ≤ Δ for all σ ∈ K , we get the following bounds, which finish the proof of our simplicial deformation theorem (Theorem 2.5.1). Remark 2.5.10. The influence of Simplicial regularity as measured by κ$_{1}$ and κ$_{2}$ is clearly revealed by the statement of our deformation theorem (Theorem 2.5.1). Explicit constants are a simple yet useful part of the result; as observed above in Remark 2.5.2, the statement of this theorem leads to an easy observation that the flat norm distance between T and P can be made a small as desired by subdividing the simplicial complex in a manner that keeps the regularity constants bounded. This can be done, for example, by using the subdivision algorithm of Edelsbrunner and Grayson [18]. 42 T − ( P + Q 2) = ∂R + Q$_{1}$, hence T − P = ∂R + Q. M( P ) ≤ (4 ϑ$_{K}$ ) $^{k}$M( T ) + Δ (4 ϑ$_{K}$ ) k $^{+1}$M( ∂T ) = (4 ϑ$_{K}$ ) k $^{(}$M( T ) + Δ (4 ϑ$_{K}$ ) M( ∂T ) $^{)}$, M( ∂P ) ≤ (4 ϑ$_{K}$ ) k $^{+1}$M( ∂T ) , M( R ) ≤ Δ M( ψ 1 $_{#}$( T )) ≤ Δ (4 ϑ$_{K}$ ) $^{k}$M( T ) , and M( Q ) ≤ Δ (4 ϑ$_{K}$ ) $^{k}$M( ∂T ) + Δ (4 ϑ$_{K}$ ) k $^{+1}$M( ∂T ) = Δ (4 ϑ$_{K}$ ) $^{k}$(1 + 4 ϑ$_{K}$ ) M( ∂T ) .