Hello again! More of the same bumbling down the road of algebraic topology. This time, I am trying to figure out exactly how much information the face poset of a CW complex encodes. It has often been remarked that a CW complex has much more data in it than a simplicial complex due to the fact that keeps track of characteristic maps; and so in a round about way this question can be thought of as an attempt to quantify exactly how much extra information there really is.
As a first observation, the nerve theorem requires that any regular CW complex must be homotopy equivalent to the geometric realization of the nerve of its face poset; and so the above question is definitely false with respect to homotopy equivalence.
However, can we do much better? Clearly this result will hold for simplicial complexes, since the face poset of a simplicial complex is an abstract simplicial complex whose realization is homeomorphic to the original simplicial complex. But does this hold for CW complexes; ie if we have two cell complexes with the same face posets, is it also true that they must automatically be homeomorphic? I am even happy if this only works in the limited case where the CW complexes are finite and the boundary of any cell is a subcomplex (ie regularity).
The underlying motivation for this question comes from a more practical application. I have a pair of fibrations over distinct cell complexes, which I would like to show are equal. To do this, I need to find some way to reduce them both to fibrations over some common base space. I can construct an isomorphism between the face posets of these cell complexes relatively easily, but I am struggling to figure out how (or even if) this extends to a homeomorphism of the cell complexes. As a result, I would take an answer to this secondary question over the topic question, but I am beginning to suspect that it may be impossible to do this in general.
Best Answer
It is not hard to reconstruct a finite regular CW complex from its face poset. This is essentially the order complex construction. One just has to understand the order complex $\Delta(P)$ of the poset $P$ not as a mere simplicial complex, but rather as the barycentric subdivision of a "cone complex" $C(P)$, where a cone in $C(P)$ is the union of the simplices of $\Delta(P)$ that correspond to all nonempty chains in the downset of a $p\in P$. The face poset of $C(P)$ is $P$ again (where by a face I mean a nonempty cone). In particular, if you start with the face poset of a regular CW complex, you get the original CW-complex.
For details see C. McCrory, Cone complexes and PL transversality (1975) and A. Bj\"orner, Posets, regular CW complexes and Bruhat order, Europ. J. Combin. 5 (1984), 7-16. (My understanding is that Bj\"orner's paper has been a standard reference for such purposes in a combinatorialist community apparently not aware of McCrory's paper.)