# Cubical approximation theorem for cubical complexes

A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.

I have found a claim in a preprint from 2008 that no such theorem is known for cubical complexes. Can anyone substantiate or disprove this claim?

• In a cubical complex, you can subdivide in splitting $n$-cubes into $2^n$ cubes in the standard way, and so forth. Is this what you call "subdivide" or you allow more flexibility in the definition of "subdividing"? (In both cases, I don't know the answer to your question.) – YCor Oct 7 '16 at 16:32
• If by "cubical complexes' you mean complexes of cubes, where etc. then the situation is simple and not truly interesting. However, sometime during 1960s I developed the theory of cubical complexes meant as subcomplexes of the full cube complex. Later, together with J.Blass, we have published a series of 5 papers in Italian Academy of Science journal around 1972-3. It contained the whole homology theory, including a homotopy property. Important: the cubical maps were induced by the projections of the full cube--a very restricted notion but sufficiently flexible for the homotopy & homology. – Włodzimierz Holsztyński Oct 7 '16 at 17:04
• Wrt the last comment, are those papers available as pdfs? Secondly, I do not have Massey's "Singular homology" available, but I recall he uses a cubical approach. Does he have a cubical approximation theorem? Finally, there are comments on related matters at mathoverflow.net/questions/3656/… – Ronnie Brown Oct 9 '16 at 9:51