# Is there a relative Pachner theorem?

Consider an $n$-dimensional PL manifold $P$ together with a closed subpolyhedron $P_0$ such that there exists a finite combinatorial triangulation of $P$ that restricts to a triangulation of $P_0$.

Is it true that any two combinatorially equivalent combinatorial triangulations of $P$ that coincide on $P_0$ are related by a finite sequence of bistellar flips (Pachner moves) which are the identity on $P_0$?

The case of $P$ a $3$-manifold with boundary $\partial P= P_0$ is Theorem 2.1.2 of Quantum Invariants of Knots and Three Manifolds. My primary motivation for this question is to try to massage such a theorem, if it exists, to try to answer a previous MO question which I asked. But I also care about the answer for its own sake. Note that $P_0$ is not a submanifold, but only a subpolyhedron. Even if the statement isn't true in full generality, is it at least true if $P_0$ is somehow not "too pathological"?

In his paper: http://arxiv.org/pdf/math/9911256.pdf Lickorish outlines a proof of theorem (Theorem 5.10) he credits to Pachner and Newman which says that two combinatorial $n$-manifolds with non-empty boundary are PL-homeomorphic if and only if they are related by a sequence of elementary shellings, inverse shellings and simplicial isomorphism.