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The integral polytope group

Florian Funke
From the journal Advances in Geometry

Abstract

We show that the Grothendieck group associated to integral polytopes in ℝn is free-abelian, by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of ℝn.

MSC 2010: 52B20

Funding statement: The author was supported by GRK 1150 ‘Homotopy and Cohomology’ funded by the DFG, the Max Planck Institute for Mathematics, and the Deutsche Telekom Stiftung.

Acknowledgements

We thank Stefan Friedl, Fabian Henneke, Dawid Kielak, and Wolfgang Lück for many fruitful discussions. We also thank the referee for carefully reading our work and for pointing out important connections to toric geometry.

  1. Communicated by: M. Joswig

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Received: 2018-01-29
Revised: 2019-02-07
Published Online: 2019-09-11
Published in Print: 2021-01-27

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