My question concerns PL-homeomorphism and shellability.


I see from [a previous question here](https://mathoverflow.net/questions/62958/do-bistellar-flips-preserve-shellability) that bistellar flips do not preserve shellability. However, the following two results are corollaries of Pachner's Theorem:


1. Every simplicial PL-sphere is the boundary of a shellable simplicial ball.
2. Bistellar equivalence implies PL-homeomorphism.


These results seem to be incompatible with the existence of non-shellable simplicial spheres, so I'm trying to find the flaw in my understanding. 


If every simplicial PL-sphere is the boundary of a shellable ball, then how can we ever obtain non-shellable objects via bistellar flips? Is every triangulated sphere the boundary of a shellable object, while not being shellable itself?



How does Pachner's Theorem preserve shellability in simplicial spheres?
My understanding is that the proof of shellability uses Pachner moves in one dimension down: if $S$ is a $d$-dimensional simplicial PL-sphere, then its boundary can be partitioned into two $(d-1)$-dimensional balls (e.g. one of the simplices is a ball itself, and all of the other simplices also form a ball), there exists a sequence of Pachner moves from one triangulated ball to the other, each individual Pachner move can be interpreted geometrically as a $d$-dimensional simplex, and the sequence of Pachner moves forms a shelling of a triangulation of $S$.


So the fact that there also exists a sequence of Pachner moves from a shellable triangulation of $S$ to some other non-shellable triangulation of $S$ is irrelevant; that sequence isn't even in the same dimension as the one used to prove shellability.


