case, the subdivision need not preserve the simplicial regularity of the ( ℓ + 1)- or higher simplices as all subsequent pushing steps will take place in lower dimensional simplices. Moreover, for a given portion of current we can use the maximum of the simplicial regularity constants of the simplices it encounters while being pushed (rather than the maximum over all simplices in the complex). For all ϵ > 0 and nonnegative integers k < p , let N ϵ k denote the set of all points in the ( k +1)-skeleton of K with positive distance less than ϵ from the k -skeleton of K (i.e., all points in the interior of the ( k + 1)-simplices of K which are close to the k -skeleton). Let T ¬ N ϵ p − 1 denote the restriction of the current T to the set N ϵ p − 1 and note that where α$_{K}$ and β are as in (the assumed true) Conjecture 3.3.4 and choose ϵ > 0 to make each of the masses in Equation (3.14) less than δ . We can apply Conjecture 3.3.4 with this ϵ to obtain a subdivision M$_{ϵ}$ of K and a subcomplex M ′ ϵ such that the portion of each of our currents which lies in M$_{ϵ}$ \ M ′ ϵ and is not already on the ( p − 1)-skeleton (so is not fixed by the first projection) has mass less than δ . This portion of each current increases in mass by a factor of at most (2 m + 2 n + ϵ ) α$_{K}$ when projecting to the ( p − 1)- skeleton (see proof of Theorem 3.3.1). Letting T ′ i denote the result of projecting T$_{i}$ to 71 lim ϵ ↓ 0 M( T$_{i}$ ¬ N ϵ p − $_{1}$) = 0 , lim ϵ ↓ 0 M( S$_{j}$ ¬ N ϵ p − $_{1}$) = 0 , lim ϵ ↓ 0 M( ∂T$_{i}$ ¬ N ϵ p − $_{1}$) = 0 , lim ϵ ↓ 0 M( ∂S$_{j}$ ¬ N ϵ p − $_{1}$) = 0 . (3.14) δ = β α$_{K}$ min 1 ≤ i ≤ m, 1 ≤ j ≤ n { M( T$_{i}$ ) , M( ∂T$_{i}$ ) , M( S$_{j}$ ) , M( ∂S$_{j}$ ) } (3.15) Let