that there are now m d -currents T$_{1}$, T$_{2}$, . . . , T$_{m}$ and n ( d + 1) -currents S$_{1}$, S$_{2}$, . . . , S$_{n}$ to push on to the complex to yield the corresponding simplicial chains P$_{i}$ and O$_{j}$ . There is a series of projection centers (as in the proof of Theorem 3.2.4 and depending on ϵ , K , the T$_{i}$ and S$_{j}$ ) which can be used with every current T$_{i}$ and S$_{j}$ to obtain the bounds: Moreover, if we let π$_{K}$ denote the projection map that uses these centers to push ( d − 1) − , d − , and ( d + 1) -currents to chains on the complex, then we have that: 64 π$_{K}$   m ∑ i =1 a$_{i}$∂T$_{i}$$^{}$  = m ∑ i =1 a$_{i}$π$_{K}$ ( ∂T$_{i}$ ) π$_{K}$   m ∑ i =1 a$_{i}$T$_{i}$ + n ∑ j =1 b$_{j}$∂S$_{j}$$^{}$  = m ∑ i =1 a$_{i}$π$_{K}$ ( T$_{i}$ ) + n ∑ j =1 b$_{j}$∂ ( π$_{K}$ ( S$_{j}$ )) π$_{K}$   n ∑ j =1 b$_{j}$S$_{j}$$^{}$  = n ∑ j =1 b$_{j}$π$_{K}$ ( S$_{j}$ ) • π$_{K}$ commutes with the boundary operator (i.e., π$_{K}$ ( ∂A ) = ∂π$_{K}$ ( A ) where A is any d - or ( d + 1) -current) • π$_{K}$ is linear on the currents T$_{i}$ , ∂T$_{i}$ , S$_{j}$ and ∂S$_{j}$ . That is, for any scalars a$_{i}$ and b$_{j}$ , M( P$_{i}$ ) ≤ ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − $^{d}$M( T$_{i}$ ) + Δ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − d $^{+1}$M( ∂T$_{i}$ ) M( ∂P$_{i}$ ) ≤ ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − d $^{+1}$M( ∂T$_{i}$ ) F ( T$_{i}$, P$_{i}$ ) ≤ Δ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − $^{d}$(M( T$_{i}$ ) + (1 + (2 m + 2 n + ϵ ) ϑ$_{K}$ ) M( ∂T$_{i}$ )) M( O$_{j}$ ) ≤ ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − d − $^{1}$M( S$_{j}$ ) + Δ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − $^{d}$M( ∂S$_{j}$ ) M( ∂O$_{j}$ ) ≤ ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − $^{d}$M( ∂S$_{j}$ ) F ( S$_{j}$, O$_{j}$ ) ≤ Δ((2 m + 2 n + ϵ ) ϑ$_{K}$ ) p − d − $^{1}$(M( S$_{j}$ ) + (1 + (2 m + 2 n + ϵ ) ϑ$_{K}$ ) M( ∂S$_{j}$ ))