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Bigraded Betti numbers of certain simple polytopes

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Abstract

The bigraded Betti numbers β −i,2j(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers β −i,2j(P) reflect the combinatorial structure of P, as well as the topological structure of the corresponding moment-angle manifold Z P ; thus, they find numerous applications in combinatorial commutative algebra and toric topology. We calculate certain bigraded Betti numbers of the type β −i,2(i+1) for associahedra and apply the calculation of bigraded Betti numbers for truncation polytopes to study the topology of their moment-angle manifolds. Presumably, for these two series of simple polytopes, the numbers β −i,2j(P) attain their minimum and maximum values among all simple polytopes P of fixed dimension with a given number of facets.

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References

  1. V. M. Bukhshtaber and T. E. Panov, Torus Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004) [in Russian].

    MATH  Google Scholar 

  2. R. P. Stanley, Progress in Mathematics, Vol. 41: Combinatorics and Commutative Algebra, 2nd ed. (Birkhäuser, Boston, MA, 1996).

    Google Scholar 

  3. T. Panov, “Cohomology of face rings, and torus actions,” in London Mathematical Society Lecture Note Series, Vol. 347: Surveys in Contemporary Mathematics (Cambridge Univ. Press, Cambridge, 2008), pp. 165–201; arXiv: math. AT/0506526.

    Google Scholar 

  4. N. Terai and T. Hibi, “Computation of Betti numbers of monomial ideals associated with stacked polytopes,” Manuscripta Math. 92(4), 447–453 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Choi and J. S. Kim, “A combinatorial proof of a formula for Betti numbers of a stacked polytope,” Electron. J. Combin. 17, Research Paper 9 (2010); arXiv: math. CO/0902.2444.

  6. F. Bosio and L. Meersseman, “Real quadrics in C n, complex manifolds and convex polytopes,” Acta Math. 197(1), 53–127 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  7. J. D. Stasheff, “Homotopy associativity of H-spaces. I,” Trans. Amer. Math. Soc. 108, 275–292 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  8. V.M. Buchstaber, “Lectures on toric topology,” Trends in Math. 10(1), 1–64 (2008).

    MathSciNet  Google Scholar 

  9. Macaulay 2, http://www.math.uiuc.edu/Macaulay2/.

  10. T. Panov, “Moment-angle manifolds and complexes,” Trends in Math. 12(1), 43–69 (2010).

    Google Scholar 

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    I. Yu. Limonchenko

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  1. I. Yu. Limonchenko
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Correspondence to I. Yu. Limonchenko.

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Original Russian Text © I. Yu. Limonchenko, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 3, pp. 373–388.

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Limonchenko, I.Y. Bigraded Betti numbers of certain simple polytopes. Math Notes 94, 351–363 (2013). https://doi.org/10.1134/S000143461309006X

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  • DOI: https://doi.org/10.1134/S000143461309006X

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