When are nontopological bistellar flips manifold-preserving? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:44:47Z http://mathoverflow.net/feeds/question/49628 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49628/when-are-nontopological-bistellar-flips-manifold-preserving When are nontopological bistellar flips manifold-preserving? manifold-destiny 2010-12-16T10:17:44Z 2010-12-22T23:31:44Z <p>A <em>topological bistellar flip</em> is the term used by <a href="http://www.springerlink.com/content/yjf3gpbej4yr71ml/" rel="nofollow">Dougherty, Faber, and Murphy</a> 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, ie, 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.</p> <p>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.</p> http://mathoverflow.net/questions/49628/when-are-nontopological-bistellar-flips-manifold-preserving/50203#50203 Answer by Sergey Melikhov for When are nontopological bistellar flips manifold-preserving? Sergey Melikhov 2010-12-22T22:49:49Z 2010-12-22T23:31:44Z <p>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</p> <blockquote> <p>on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.</p> </blockquote> <p>is supposed to be about trying to apply a bistellar move to the <em>boundary</em> 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. </p> <p>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 <a href="http://www.ams.org/mathscinet-getitem?mr=2012938" rel="nofollow">this review</a>, 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".</p>