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A topological bistellar flip is the term used by Dougherty, Faber, and Murphy to describe a bistellar flip that does not cause any face of a complex to be duplicated. Suppose we consider a generalized notion of bistellar flips that are non-topological, i.e., they can duplicate existing faces of complexes. For instance, on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.

Is this notion already defined in the literature? Which classes of nontopological bistellar flips are known to preserve the type of a manifold? It seems likely to me that in low dimensions a good deal may already be known.

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    $\begingroup$ Bistellar flips (that is, Pachner moves) do not change the topology of the manifold, see en.wikipedia.org/wiki/Pachner_moves $\endgroup$
    – Bruno Martelli
    Commented Dec 16, 2010 at 10:29
  • $\begingroup$ Perhaps I am describing the wrong operation. Consider a 3-simplex to which we apply a (2,2) bistellar flip. The operation drags several edges together. This would seem to collapse a 2-sphere into a 1-sphere. $\endgroup$
    – manifold-destiny
    Commented Dec 16, 2010 at 19:36
  • $\begingroup$ Could you define what you mean by a non-topological (2,2) bistellar flip? What does "manifold-preserving" mean? $\endgroup$
    – Ryan Budney
    Commented Dec 22, 2010 at 0:42

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By a topological (2,2)-flip Dougherty, Faber, and Murphy mean a bistellar move on 2-manifolds (and not on 3-manifolds). So it looks like the example

on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.

is supposed to be about trying to apply a bistellar move to the boundary of a 3-simplex. It further looks like the geometers want to see the result of such a generalized bistellar flip as the 2-disk, triangulated by the suspension over the 1-simplex. As a topologist, I'm deeply troubled by such vision of a 2-dimensional Pachner move (e.g. what of a "flip" does it retain if so generalized?) I would either think of this as a 3-dimensional move indeed (see more about this below) or else I would consider the following to be the result of this move: two copies of the suspension over a 1-simplex, glued along their boundaries. This is no longer a simplicial complex, but a "pseudo-complex" in the sense of the Hilton-Wylie textbook, and a "singular triangulation" of a more modern tradition.

Certain generalizations of bistellar moves to singular triangulations have been studied by Matveev and his students; they are precisely dual to Matveev's moves on special spines (concerning the duality, see this review, though there must be better references). Some of these generalized bistellar moves are not supported by homeomorphisms (so maybe geometers would call them "non-topological"). For instance, there is a move that collapses the join of a 1-simplex and S, where the 1-sphere S is the union of two copies of a 1-simplex along their boundaries, onto the suspension over a 1-simplex. This 3-dimensional singular flip can be decomposed into a sequence of two "(2,2) non-topological flips".

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