Barycentric Subdivisions of Convex Complexes are Collapsible

Karim Adiprasito; Bruno Benedetti

doi:10.1007/s00454-019-00137-3

Barycentric Subdivisions of Convex Complexes are Collapsible

Karim Adiprasito^*
, Bruno Benedetti

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d≤ 3. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in R^d is simplicially collapsible after d- 2 derived subdivisions at most. This presents progress on an old question by Goodrick.

Original language	English
Pages (from-to)	608-626
Number of pages	19
Journal	Discrete and Computational Geometry
Volume	64
Issue number	3
DOIs	https://doi.org/10.1007/s00454-019-00137-3
State	Published - 1 Oct 2020

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© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

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abstract = "A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d≤ 3. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in Rd is simplicially collapsible after d- 2 derived subdivisions at most. This presents progress on an old question by Goodrick.",

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N2 - A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d≤ 3. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in Rd is simplicially collapsible after d- 2 derived subdivisions at most. This presents progress on an old question by Goodrick.

AB - A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d≤ 3. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in Rd is simplicially collapsible after d- 2 derived subdivisions at most. This presents progress on an old question by Goodrick.

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Barycentric Subdivisions of Convex Complexes are Collapsible

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