A point set whose space of triangulations is disconnected

Santos, Francisco

doi:10.1090/S0894-0347-00-00330-1

A point set whose space of triangulations is disconnected

Author: Francisco Santos
Journal: J. Amer. Math. Soc. 13 (2000), 611-637
MSC (2000): Primary 52B11; Secondary 52B20
DOI: https://doi.org/10.1090/S0894-0347-00-00330-1
Published electronically: March 29, 2000
MathSciNet review: 1758756
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Abstract:

By the ``space of triangulations" of a finite point configuration $\mathcal{A}$ we mean either of the following two objects: the graph of triangulations of $\mathcal{A}$ , whose vertices are the triangulations of $\mathcal{A}$ and whose edges are the geometric bistellar operations between them or the partially ordered set (poset) of all polyhedral subdivisions of $\mathcal{A}$ ordered by coherent refinement. The latter is a modification of the more usual Baues poset of $\mathcal{A}$ . It is explicitly introduced here for the first time and is of special interest in the theory of toric varieties.

We construct an integer point configuration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This triangulation is an isolated point in both the graph and the poset, which proves for the first time that these two objects cannot be connected.

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