Barycentric Subdivisions of Convex Complexes are Collapsible

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 \(d \le 3\). 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 \(\mathbb {R}^d\) is simplicially collapsible after \(d-2\) derived subdivisions at most. This presents progress on an old question by Goodrick.

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Notes

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    This is true because any movement of star-center x normal to the geodesic span of \(\tau \) and x induces a motion of the hemisphere \( \mathrm {N}^1_{(u,\tau )}B_u\) in \(\mathrm {N}^1_{(u,\tau )} S^d\), specifically, by moving the midpoint of the hemisphere in the same direction.

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Acknowledgements

Karim Adiprasito acknowledges the support by ISF Grant 1050/16 and ERC StG 716424-CASe. Bruno Benedetti acknowledges the support by an NSF Grant 1600741, the DFG Collaborative Research Center TRR109, and the Swedish Research Council VR 2011-980. Part of this work was supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

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Correspondence to Karim Adiprasito.

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Dedicated to the memory of Ricky Pollack.

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Adiprasito, K., Benedetti, B. Barycentric Subdivisions of Convex Complexes are Collapsible. Discrete Comput Geom 64, 608–626 (2020). https://doi.org/10.1007/s00454-019-00137-3

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