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If $P$ and $Q$ are compact codimension zero submanifolds of a PL manifold, say that they meet nicely if $P\cap Q$ is a codimension zero submanifold of both $\partial P$ and $\partial Q$. In particular then $P\cup Q$ is a codimension zero submanifold.

Question: Given a triangulation of a compact PL manifold $M$, do there always exist compact codimension zero PL submanifolds $M_1,\dots ,M_k$ of $M$ such that:

(1) $M=M_1\cup\dots\cup M_k$,

(2) for every $i$, $M_1\cup\dots\cup M_{i-1}$ meets $M_i$ nicely, and

(3) for every $i$, $M_i$ is contained in some simplex of the given triangulation of $M$.

[Note: These submanifolds are not going to be subcomplexes for the given triangulation. They will be subcomplexes for some finer trangulation. But it is required that each $M_i$ is a subset of some original simplex.]

[EDIT: I do have a strategy for trying to prove that this. I am thinking of induction on the dimension of $M$, using the handle structure associated with the second barycentric subdivision. But for the induction it would require proving a stronger statement in the case when $M$ is PL homeomorphic to a disk. Even if this works, I'd still be interested in a reference if anyone has one.]

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    $\begingroup$ It looks like an impossible demand to me. If the $M_i$'s are contained in simplices of a fixed triangulation they would have to be simplices, but then it would be impossible for them to intersect in co-dimension zero manifolds. Maybe this isn't exactly the question you intended? $\endgroup$ Mar 6, 2014 at 2:43
  • $\begingroup$ Okay, the question makes sense to me now. So if we were to take the $M_i$'s to simply be all the top-dimensional simplices, the issue that fails (potentially) is the niceness criterion, specifically (2). You could get possible intersections of higher co-dimension. $\endgroup$ Mar 6, 2014 at 4:01
  • $\begingroup$ The PL triangulations literature isn't very easy to search. There's a bunch of Germans that might have quick references for you but as far as I know they don't contribute here. Udo Pachner or Ulrich Pinkall might be good people to ask. Maybe Ed Schwartz at Cornell? $\endgroup$ Mar 6, 2014 at 4:26

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