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Small $ f$-vectors of 3-spheres and of 4-polytopes


Authors: Philip Brinkmann and Günter M. Ziegler
Journal: Math. Comp. 87 (2018), 2955-2975
MSC (2010): Primary 52B11, 52B55; Secondary 52C40
DOI: https://doi.org/10.1090/mcom/3300
Published electronically: February 14, 2018
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Abstract: We present a new algorithmic approach that can be used to determine whether a given quadruple $ (f_0,f_1,f_2,f_3)$ is the $ f$-vector of any convex $ 4$-dimensional polytope, or more generally of a strongly regular cellular $ 3$-sphere, that is, a regular cell complex homeomorphic to the $ 3$-dimensional sphere such that any intersection of two faces (cells) is a face.

By implementing this approach, we classify the $ f$-vectors of $ 4$-polytopes in the range $ f_0+f_3\le 22$.

In particular, we prove that there are $ f$-vectors of strongly regular cellular $ 3$-spheres that are not $ f$-vectors of any convex $ 4$-polytopes. This answers a question that may be traced back to the works of Steinitz (1906/1922). In the range $ f_0+f_3\le 22$, there are exactly three such $ f$-vectors with $ f_0\le f_3$, namely $ (10,32,33,11)$, $ (10,33,35,12)$, and $ (11,35,35,11)$.


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Additional Information

Philip Brinkmann
Affiliation: Institut für Mathematik, FU Berlin Arnimallee 2 14195 Berlin, Germany
Email: webmaster@phi-fotos.de

Günter M. Ziegler
Affiliation: Institut für Mathematik, FU Berlin Arnimallee 2 14195 Berlin, Germany
Email: ziegler@math.fu-berlin.de

DOI: https://doi.org/10.1090/mcom/3300
Received by editor(s): October 19, 2016
Received by editor(s) in revised form: March 26, 2017, and May 27, 2017
Published electronically: February 14, 2018
Additional Notes: The first author was funded by DFG through the RTG Methods for Discrete Structures.
The second author was supported by DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.
Article copyright: © Copyright 2018 American Mathematical Society

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