Suppose we had a CW-complex $X$ with decomposition $X_{i}$
Is its face poset, consisting of cells and covers generated by the attachment of cells to one another, an Eulerian poset? What would be the most concise method of proving this?
If not can we prove that; there are at most two $y$ such that $y$ covers $x$ and $z$ covers $y$?
EDIT: Can we ensure that either of these conditions by restricting to regular CW-complexes?