Subdivisions, shellability, and collapsibility of products

Karim A. Adiprasito*, Bruno Benedetti

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We prove that the second derived subdivision of any rectilinear triangulation of any convex polytope is shellable. Also, we prove that the first derived subdivision of every rectilinear triangulation of any convex 3-dimensional polytope is shellable. This complements Mary Ellen Rudin's classical example of a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is a new relative notion of shellability that characterizes the behavior of shellable complexes under gluing. As a corollary, we obtain a new characterization of the PL property in terms of shellability: A triangulation of a sphere or of a ball is PL if and only if it becomes shellable after sufficiently many derived subdivisions. This improves on PL approximation theorems by Whitehead, Zeeman and Glaser, and answers a question by Billera and Swartz. We also show that any contractible complex can be made collapsible by repeatedly taking products with an interval. This strengthens results by Dierker and Lickorish, and resolves a conjecture of Oliver. Finally, we give an example that this behavior extends to non-evasiveness, thereby answering a question of Welker.

Original languageAmerican English
Pages (from-to)1-30
Number of pages30
Issue number1
StatePublished - 1 Feb 2017

Bibliographical note

Publisher Copyright:
© 2016, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.


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