Question: What is the first appearance in the literature of one of the following statements:

The result of intersecting a simplex with a cell of the dual subdivision is a cube

There is a coherent way of defining a subdivision of a simplex into cubes which is intermediate between the trivial and the barycentric subdivision.

The first statement appears naturally in geometric topology, since the dual subdivision was introduced by Poincare in his first addendum to Analysis Situs in order to prove the equality of "complementary" Betti numbers. A picture of the 3-dimensional case appears here. Denis Sullivan has a name for this: the intersection of simplices with their duals gives a cubical subdivision which he calls the *pair subdivision*.

I would have expected that the second statement is part of the proof of the equivalence between simplicial and cubical approaches to homotopy theory, but I do not know this part of the literature. It basically implies that the barycentric subdivision map on singular chains factors through a map to cubical chains

Here's what I have managed so far:

- 1899: I could not find such a statement in the first two addenda to Analysis Situs, though I could have missed it because of archaic language.
- 1953: After Serre used cubical homology to construct the spectral sequence of a fibration, Eilenberg and MacLane used acyclic models to prove the equivalence with singular homology. At the end of their paper, they write down an explicit map between the two chain complexes, but it is not the one above.
- 1986: Phillips and Stone used this subdivision in dimension 4. (This was pointed out to me by Sullivan)
- 1992: Shtanʹko and Stogrin give a short proof that the dual subdivision of a simplex is a cubulation. (This reference was pointed out by an anonymous referee)