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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: 

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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: 

• 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$_{j}$ , 

<img src='content_image/316342.jpg'>

