Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $P$.

Is it true that there is a dual cell structure, also a regular CW complex, and whose face poset is the opposite (dual) poset $P^{\mbox{op}}$?

So far I can't find a reference for this in the literature. One often starts with a *triangulation* of $M$, and I am looking for a more general statement.

UPDATE: I accepted the answer to the original question, which points out that this is not true in the generality stated. But I would still like to understand under what conditions something like this holds. Suppose instead that instead of a regular CW-complex structure on a compact manifold, we have a polyhedral (convex cell) complex which is a PL-sphere. Under these conditions, is there a dual polyhedral structure with opposite poset as above?

triangulationthis is a very special property: the triangulation must be combinatorial (i.e., give rise to a $PL$-structure). In general, links of simplices may even have wrong homotopy type. $\endgroup$