choose centers to retract from in each simplex σ such that the expansion of mass of the current restricted to that simplex is bounded by 4 ϑ$_{σ}$ / V$_{ℓ}$( B $_{σ}$). The bound specified in the Lemma follows when we consider retracting the entire current over multiple simplices in K , and set ϑ$_{K}$ = max$_{σ}$$_{∈}$$_{K}$ ϑ$_{σ}$ as the generic upper bound that holds for all simplices in K . 

Bound on complete sequence of retractions. We can apply the bound specified in Lemma 2.5.9 over multiple levels ℓ . Pushing T onto the d -skeleton of p -complex K multiplies the mass of T by a factor of at most (4 ϑ$_{K}$ ) $^{k}$. Likewise, pushing ∂T on to the ( d − 1)-skeleton multiplies the mass of ∂T by a factor of at most (4 ϑ$_{K}$ ) k $^{+1}$. 

## Bounding the distance between the current and its simplicial approximation 

In the final step, we construct the simplicial current P approximating the original cur- rent T , and bound the flat norm distance between the two. Since we are now considering retraction maps over many simplices simultaneously, we let φ$_{i}$ denote the global projec- tion from the ( p − i + 1) − skeleton to the ( p − i ) − skeleton, suppressing the particular x and a . We denote the composition of all these steps as ψ 1 ≡ φ$_{k}$ ◦ · · · ◦ φ$_{1}$ and hence we map T forward by ψ $^{1}$, picking centers (see Lemma 2.5.8) to project from in each step and in each simplex. We pick each of these centers such that the retractions map ∂T with bounded amplification of mass as well (see Lemma 2.5.9). 

The homotopy formula [29, Section 7.4.3] states that given a smooth homotopy g from f$_{0}$ to f$_{1}$ where f$_{0}$, f$_{1}$ : U ⊆ R n$_{0}$ → R n$_{1}$ are smooth functions with g (0 , x ) = f$_{0}$ ( x ) and g (1 , x ) = f$_{1}$ ( x ), if T is a d -current and f − $^{1}$( F ) ∩ spt f is compact for every compact set F ⊆ R $^{n$_{1}$}$, we have that the difference in pushforwards of T under f$_{1}$ and f$_{0}$ is given 

