chain equivalence between cubical chain complex and simplicial chain complex I remember hearing before that for a topological space, the cubical chain complex and the simplicial chain complex are chain equivalent. Is this true? If yes, can someone provide me with a reference where I could see how this chain equivalence is constructed? A natural choice /guess would be the chain map going from the cubical chain complex to the simplicial one (because cubes could be dissected into simplicies), however i have no clue about the reverse chain map. Thank you

This equivalence was first proved in the classical paper

Eilenberg, S., and Mac Lane, S., Acyclic models. Amer. J. Math. 75 (1953) 189–199.

The method of acyclic models was extended from chain complexes to crossed complexes in Section 10.4 of the book Nonabelian Algebraic Topology. The advantage of the more general case is that it takes into account the fundamental groups and groupoids and their actions and so more readily yields weak equivalences of spaces.