My primary question is: given a cellular decomposition of a sphere is there any way to check if it can be embedded as the boundary of a polytope? 


My question is motivated by the following problem. I began with a polyhedral cell complex $P$ homeomorphic to a ball. Then I had an unbounded polyhedral cone $C$ of the same dimension as $P$. I made the cone homeomorphic to a ball by taking its intersection with a ball sufficiently large so that all the bounded portions of the cone were not changed. Call this new polyhedral ball $C^\*$. Then I defined a continuous mapping from $\partial C^\*$ to $\partial P$ and used the pasting lemma to define my cellular decomposition of the resulting sphere. 


Since I am beginning with two polyhedral objects I would like to end up with something polyhedral, in this case a polytope. My method seems equivalent to "blocking" the unbounded portion of the cone with $P$, which seems should remain polyhedral.



Cellular decomposition of a sphere to polytope
The "primary question" is a well-known hard problem, and I think this reference:


MR0889977 (89b:52009) Reviewed 
Bokowski, Jürgen(D-DARM); Sturmfels, Bernd(D-DARM)
Polytopal and nonpolytopal spheres: an algorithmic approach. 
Israel J. Math. 57 (1987), no. 3, 257–271. 
52A25 (05B35) 
PDF Clipboard Journal Article Make Link


Is still close to the last word 25 years later (maybe @GilKalai can correct me).


