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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
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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Additional Information

Francisco Santos
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39005, Santander, Spain
Email: santos@matesco.unican.es

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00330-1
PII: S 0894-0347(00)00330-1
Keywords: Triangulation, point configuration, bistellar flip, polyhedral subdivision, Baues problem
Received by editor(s): August 3, 1999
Received by editor(s) in revised form: March 6, 2000
Published electronically: March 29, 2000
Additional Notes: This research was partially supported by grant PB97–0358 of the Spanish Dirección General de Enseñanza Superior e Investigación Científica.
Article copyright: © Copyright 2000 American Mathematical Society