<Verb>DiagonalApproximation(X):: RegCWComplex --> RegCWMap, RegCWMap</Verb><P/>

<P/> Inputs a regular CW-complex <M>X</M> and outputs a pair <M>[p,\iota]</M> 
of maps of CW-complexes. The map <M>p\colon X^\Delta \rightarrow X</M>
 will often
be a homotopy equivalence. This is always the case if <M>X</M> is the CW-space 
of any pure cubical complex. In general, one can test to see if the induced 
chain map <M>p_\ast \colon C_\ast(X^\Delta) \rightarrow C_\ast(X)</M> is an 
isomorphism on integral homology. The second map  
<M>\iota \colon X^\Delta \hookrightarrow X\times X</M> is an inclusion into the direct product. 
If <M>p_\ast</M> induces an isomorphism on homology then the chain map
<M>\iota_\ast\colon C_\ast(X^\Delta) \rightarrow C_\ast(X\times X)</M> can be used to compute the cup product.


