CochainComplex(K):: CubicalComplex --> CochainComplex
CochainComplex(K):: PureCubicalComplex --> CochainComplex
CochainComplex(K):: PurePermComplex --> CochainComplex
CochainComplex(Y):: RegCWComplex --> CochainComplex
CochainComplex(K):: SimplicialComplex --> CohainComplex

Inputs a cubical, or pure cubical, or pure permutahedral or simplicial complex K and returns its cochain complex of free abelian groups. In degree n this cochain complex has one free generator for each n-dimensional cell of K.

Inputs a regular CW-complex Y and returns a cochain complex C which is chain homotopy equivalent to the cellular cochain complex of Y. In degree n the free abelian cochain group C_n has one free generator for each critical n-dimensional cell of Y with respect to some discrete vector field on Y.