# Dual cell structures on manifolds

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?

• As far as I know, even with a triangulation this 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. – Alex Degtyarev Jan 19 '15 at 21:50

This text gives some details explaining that even with triangulations this may not work: a construction of a non-combinatorial triangulation (as double suspension of a homology sphere) and further references.

• Just to add a bit of info: The text linked is a 3-page undated document, "The Triangulation Conjecture," by Ciprian Manolescu. – Joseph O'Rourke Jan 20 '15 at 1:14

In "Combinatorial cell complexes and Poincare duality", T. Basak defines a combinatorial cell complex (c.c.c.), which seem to meet your "polyhedral (convex cell) complex" description. For a c.c.c. $S$, the author defines the opposite c.c.c. $S^\circ$ as the reversal of the partial order. This construction of $S^\circ$ appears to yield the "dual polyhedral structure with opposite poset" you desire.

Shouldn't the following construction work for a cell complex K that represents a PL n-manifold? Take the barycentric subdivision K' of K. For each vertex v in K, the dual n-cell will be the union of the simplices in K' containing v. Since the link of v in K was a sphere, this union will be a ball.