Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

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A point set whose space of triangulations is disconnected
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Francisco Santos, Universidad de Cantabria**

Let
be a finite point set in the Euclidean space of a certain
dimension.
By the ``space of triangulations" of
we mean either of the
following two objects: the poset of all polyhedral subdivisions of (the so-called *Baues poset* of ,
see e.g. lecture 9 in [7]) considered as
an abstract simplicial
complex in the standard way [1]; or the *graph of
triangulations* of ,
whose
vertices are the triangulations of
and whose edges are the geometric
bistellar
operations between (*``flips''*) them, see e.g. [3].

We construct a point configuration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This means that this triangulation is an isolated element in both the Baues poset and the graph of triangulations of . The point configuration is the product of two point configurations of dimension four with 81 points and of dimension two with four elements. The triangulation is a refinement of the product of respective triangulations of and , based on the idea of ``staircase triangulations" of the product of two simplices, see [4] page 282.

Our example answers in the negative the generalized Baues question for triangulations, which asked whether the Baues poset of every point configuration has the homotopy type of a sphere of a certain dimension (see [5]). It also solves one of the ``challenges" in [8] and has potential impact not only in discrete and computational geometry but also in algebraic geometry, via the close relation between subdivisions of polytopes and toric varieties (see, e.g. [2]).

The construction is based in several respects (some obvious, some not) on the paper [6]. A draft containing all the details and proofs is available upon request to the author.

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