ChainMap(X,A,Y,B):: PureCubicalComplex, PureCubicalComplex, PureCubicalComplex, PureCubicalComplex --> ChainMap
ChainMap(f):: RegCWMap --> ChainMap
ChainMap(f):: SimplicialMap --> ChainComplex

Inputs a pure cubical complex Y and pure cubical sucomplexes X\subset Y, B\subset Y,A\subset B. It returns the induced chain map f_\ast\colon C_\ast(X/A) \rightarrow C_\ast(Y/B) of cellular chain complexes of pairs. (Typlically one takes A and B to be empty or contractible subspaces, in which case C_\ast(X/A) \simeq C_\ast(X), C_\ast(Y/B) \simeq C_\ast(Y).)

Inputs a map f\colon X \rightarrow Y between two regular CW-complexes X,Y and returns an induced chain map f_\ast\colon C_\ast(X) \rightarrow C_\ast(Y) where C_\ast(X), C_\ast(Y) are chain homotopic to (but usually smaller than) the cellular chain complexes of X, Y.

Inputs a map f\colon X \rightarrow Y between two simplicial complexes X,Y and returns the induced chain map f_\ast\colon C_\ast(X) \rightarrow C_\ast(Y) of cellular chain complexes.