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A Subexponential Size Triangulation of ℝPn

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Abstract

We break the exponential barrier for triangulations of the real projective space, constructing a trianglation of ℝPn with \({e^{\left({{1 \over 2} + o(1)} \right)\sqrt n \log n}}\) vertices.

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References

  1. K. Adiprasito: Combinatorial Lefschetz theorems beyond positivity, 2018, preprint, arXiv:1812.10454.

  2. K. Adiprasito and G. Yashfe: The Partition Complex: an invitation to combinatorial commutative algebra, 2020, Surveys in Combinatorics, British Combinatorial Committee, arXiv:2008.01044.

  3. P. Arnoux and A. Marin: The Kühnel triangulation of the complex projective plane from the view point of complex crystallography. II, Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), 167–244.

    MathSciNet  MATH  Google Scholar 

  4. A. Barvinok: A bound for the number of vertices of a polytope with applications, Combinatorica 33 (2013), 1–10.

    Article  MathSciNet  Google Scholar 

  5. B. Basak and S. Sarkar: Equilibrium and equivariant triangulations of some small covers with minimum number of vertices., J. Ramanujan Math. Soc. 30 (2015), 29–50.

    MathSciNet  MATH  Google Scholar 

  6. O. Cornea, G. Lupton, J. Oprea and D. Tanre: Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003.

    Book  Google Scholar 

  7. B. Delaunay: Sur la sphère vide, in: Proceedings Congress Toronto 1, 695–700 1928.

  8. P. Frankl, J. Pach and D. Pálvölgyi: Exchange properties of finite set-systems, 2021, preprint, arXiv:2103.14358.

  9. W. Kuhnel: Tight Polyhedral Submanifolds and Tight Triangulations, Lecture Notes in Mathematics, vol. 1612, Springer-Verlag, Berlin, 1995.

    Book  Google Scholar 

  10. F. H. Lutz: Triangulated manifolds with few vertices and vertex-transitive group actions, Berichte aus der Mathematik. [Reports from Mathematics], Verlag Shaker, Aachen, 1999, Dissertation, Technischen Universität Berlin, Berlin, 1999.

    MATH  Google Scholar 

  11. S. Murai: Tight combinatorial manifolds and graded Betti numbers., Collect. Math. 66 (2015), 367–386.

    Article  MathSciNet  Google Scholar 

  12. L. Venturello and H. Zheng: A new family of triangulations of ℝPd, Combinatorica 41 (2021), 127–146.

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We wish to thank Wolfgang Kühnel and Alexis Marin for very extensive and helpful remarks on the first version of the paper and anonymous referees for their comments and their suggestion of how to simplify the last part of the proof of Claim 3.3. We also wish to thank Arseniy Akopyan, Xavier Goaoc, Andrey Kupavskii, Gaku Liu, János Pach and Dömötör Pálvölgyi for helpful discussions.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Karim Adiprasito

  2. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Karim Adiprasito

  3. Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994

    Roman Karasev

  4. Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia, 141700

    Roman Karasev

  5. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Sergey Avvakumov

Authors
  1. Karim Adiprasito
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  2. Sergey Avvakumov
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  3. Roman Karasev
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Corresponding author

Correspondence to Karim Adiprasito.

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Dedicated to Wolfgang Kühnel on the occasion of his 70th birthday.

Has received funding from the European Research Council under the European Unions Seventh Framework Programme ERC Grant agreement ERC StG 716424 — CASe and the Israel Science Founda-tion under ISF Grant 1050/16.

Has received funding from the European Research Council under the European Unions Seventh Framework Programme ERC Grant agreement ERC StG 716424 — CASe.

Supported by the Federal professorship program grant 1.456.2016/1.4 and the Russian Foundation for Basic Research grants 18-01-00036 and 19-01-00169.

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Adiprasito, K., Avvakumov, S. & Karasev, R. A Subexponential Size Triangulation of ℝPn. Combinatorica 42, 1–8 (2022). https://doi.org/10.1007/s00493-021-4602-x

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  • DOI: https://doi.org/10.1007/s00493-021-4602-x

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