# Derived subdivisions make every PL sphere polytopal

@article{Adiprasito2013DerivedSM, title={Derived subdivisions make every PL sphere polytopal}, author={Karim A. Adiprasito and Ivan Izmestiev}, journal={Israel Journal of Mathematics}, year={2013}, volume={208}, pages={443-450} }

We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication).

#### 11 Citations

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#### References

SHOWING 1-10 OF 22 REFERENCES

Simplifying triangulations of S3

- Mathematics
- 2003

In this paper we describe a procedure to simplify any given triangulation of S 3 using Pachner moves. We obtain an explicit exponential-type bound on the number of Pachner moves needed for this… Expand

Unshellable Triangulations of Spheres

- Computer Science, Mathematics
- Eur. J. Comb.
- 1991

A direct proof is given of the existence of non-shellable triangulations of spheres which, in higher dimensions, yields new examples of such triangulations.

Geometric bistellar flips. The setting, the context and a construction

- Mathematics
- 2006

We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known… Expand

INFINITESIMAL RIGIDITY OF POLYHEDRA WITH VERTICES IN CONVEX POSITION

- Mathematics
- 2010

Let P C R 3 be a polyhedron. It was conjectured that if P is weakly convex (that is, its vertices lie on the boundary of a strictly convex domain) and decomposable (that is, P can be triangulated… Expand

Non-connected toric Hilbert schemes

- Mathematics
- 2002

Abstract.We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math.… Expand

Birational Geometry of Toric Varieties

- Mathematics
- 2002

This chapter is intended as a coffee break after the previous thirteen chapters of hard work. We will just play around with the tone varieties and see all the ingredients of the Mori program at work… Expand

Piecewise Linear Topology

- Mathematics
- 2001

The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic accounts of the subject may be foundin [ 2… Expand

Lectures on Polytopes

- Mathematics
- 1994

Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward… Expand

Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten

- Mathematics
- 1987

1. Einleitung: Das Studium der Geometrie der Simplizialkomplexe hat eine lange und reiche Tradition. Obwohl dieses Gebiet durchaus genfigend Leben aus der Kraft und Sch6nheit seiner Strukturaussagen… Expand