content_image/307713.jpg content_image/307714.jpg
Equation (2.19) gives ψ 1 $_{#}$( T ) = P + Q 2. Defining Q = Q$_{1}$ + Q$_{2}$ , Equation (2.18) gives 

<img src='content_image/307713.jpg'>

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). 

<img src='content_image/307714.jpg'>

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]. 

