ChainComplexEquivalence(X):: RegCWComplex --> List

Inputs a regular CW-complex X and returns a pair [f_\ast, g_\ast] of chain maps f_\ast\colon C_\ast(X) \rightarrow D_\ast(X), g_\ast\colon D_\ast(X) \rightarrow C_\ast(X). Here C_\ast(X) is the standard cellular chain complex of X with one free generator for each cell in X. The chain complex D_\ast(X) is a typically smaller chain complex arising from a discrete vector field on X. The chain maps f_\ast, g_\ast are chain homotopy equivalences.