with constants independent of the complex and currents involved if we subdivide the complex by means of Conjecture 3.3.4 (the subdivision does depend on the currents and complex, of course). Theorem 3.3.6. Suppose we have integers d < s ≤ q and that Conjecture 3.3.4 holds for the given q and any p such that d − 1 ≤ p ≤ s (that is, suppose we can isolate the irregularities of any p -complex in R q by suitable subdivision). Given a s -dimensional simplicial complex K in R q and a set of d -currents T$_{1}$, . . . , T$_{m}$ and ( d + 1) -currents S$_{1}$, . . . , S$_{n}$ in the underlying space of K with d < s , there exists a complex K ′ which is a subdivision of K such that we have all of the conclusions of Theorem 3.3.1 (i.e., mass and flat norm bounds and linear projection of the T$_{i}$ and S$_{j}$ to K $^{′}$) except the simplicial irregularity constant ϑ$_{K}$ ′ in the various bounds can be replaced with a constant L which does not depend on K . Proof. In the simplicial deformation theorems, the current is projected step-by-step to lower dimensional skeletons (e.g., a d -current is projected from the initial p -complex to the ( p − 1)-skeleton, then the ( p − 2)-skeleton eventually down to the d -skeleton with one more step to push the current’s boundary to the ( d − 1)-skeleton) with each projection being done by picking a center in each simplex and using it to project outward to the boundary of the simplex. The simplicial regularity constant is used to bound the expansion of mass at each projection step and is defined by Equation (3.8), a bound on the regularity of all simplices in the complex. However, this is a bit stronger than required as the projection is a local operation and the bound at each step depends only on the simplicial regularity of the simplex in question. In addition, there is no reason in principle that we cannot subdivide the complex in between steps. That is, after pushing to the ℓ -skeleton, we can further subdivide the complex and then push to the newly refined ( ℓ − 1)-skeleton. In this 70