Abstract
In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres \(\mathbb S^i \times \mathbb S^j\) with \(j \ge i\) is tight if and only if it has exactly \(i+2j+4\) vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when \(j>2i\) and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.
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Avramov, L.L., Conca, A., Iyengar, S.B.: Subadditivity of syzygies of Koszul algebras. Math. Ann. 361, 511–534 (2015)
Bagchi, B.: The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds. arXiv:1405.5675
Bagchi, B.: A tightness criterion for homology manifolds with or without boundary. Eur. J. Comb. 46, 10–15 (2015)
Bagchi, B., Datta, B.: On stellated spheres and a tightness criterion for combinatorial manifolds. Eur. J. Comb. 36, 294–313 (2014)
Barnette, D.W.: The minimum number of vertices of a simple polytope. Israel J. Math. 10, 121–125 (1971)
Barnette, D.W.: A proof of the lower bound conjecture for convex polytopes. Pacific J. Math. 46, 349–354 (1973)
Bigatti, A.: Upper bounds for the Betti numbers of a given Hilbert function. Commun. Algebra 21, 2317–2334 (1993)
Brehm, U., Kühnel, W.: Combinatorial manifolds with few vertices. Topology 26, 465–473 (1987)
Bruns, W., Herzog, J.: Cohen-Macaulay rings, Revised edn. Cambridge University Press, Cambridge (1993)
Burton, B.A., Datta, B., Singh, N., Spreer, J.: Separation index of graphs and stacked 2-spheres. arXiv:1403.5862
Datta, B., Murai, S.: On stacked triangulated manifolds. arXiv:1407.6767
Datta, B., Singh, N.: An infinite family of tight triangulations of manifolds. J. Comb. Theory Ser. A 120, 2148–2163 (2013)
Effenberger, F.: Stacked polytopes and tight triangulations of manifolds. J. Comb. Theory Ser. A 118, 1843–1862 (2011)
Eliahou, S., Kervaire, M.: Minimal resolutions of some monomial ideals. J. Algebra 129, 1–25 (1990)
Fernández-Ramos, O., Gimenez, P.: Regularity \(3\) in edge ideals associated to bipartite graphs. J. Algebraic Comb. 39, 919–937 (2014)
Flores, A.: Über \(n\)-dimensionale Komplexe, die im \(R_{2n+1}\) absolut selbstverschlungen sind. Ergeb. Math. Kolloq. 6, 4–7 (1933/34)
Fogelsanger, A.: The generic rigidity of minimal cycles. Ph.D. Thesis, Cornell University (1988)
Herzog, J., Hibi, T.: Monomial ideals. Graduate Texts in Mathematics. Springer, London (2011)
Herzog, J., Srinivasan, H.: A note on the subadditivity problem for maximal shifts in free resolutions. MSRI Proc. (to appear). arXiv:1303.6214
Hulett, H.A.: Maximum Betti numbers of homogeneous ideals with a given Hilbert function. Commun. Algebra 21, 2335–2350 (1993)
Kalai, G.: Rigidity and the lower bound theorem. I. Invent. Math. 88, 125–151 (1987)
Kühnel, W.: Tight polyhedral submanifolds and tight triangulations. Lecture Notes in Math, vol. 1612. Springer, Berlin (1995)
Kühnel, W., Lutz, F.H.: A census of tight triangulations. In: Discrete Geometry and Rigidity, Budapest, 1999, Period. Math. Hungar, vol. 39, pp. 161–183 (1999)
Lutz, F.H.: Triangulated manifolds with few vertices: combinatorial manifolds. arXiv:math.0506372
Lutz, F.H., Sulanke, T., Swartz, E.: \(f\)-vectors of \(3\)-manifolds. Electron. J. Comb. 16, 33 (2009) (Research paper R13)
Macaulay, F.S.: Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. 26, 531–555 (1927)
Migliore, J., Nagel, U.: Reduced arithmetically Gorenstein* schemes and simplicial polytopes with maximal Betti numbers. Adv. Math. 180, 1–63 (2003)
Murai, S., Nevo, E.: On the generalized lower bound conjecture for polytopes and spheres. Acta Math. 210, 185–202 (2013)
Murai, S., Nevo, E.: On \(r\)-stacked triangulated manifolds. J. Algebraic Comb. 39, 373–388 (2014)
Novik, I., Swartz, E.: Socles of Buchsbaum modules, complexes and posets. Adv. Math. 222, 2059–2084 (2009)
Novik, I., Swartz, E.: Applications of Klee’s Dehn-Sommerville relations. Discrete Comput. Geom. 42, 261–276 (2009)
Novik, I., Swartz, E.: Face numbers of pseudomanifolds with isolated singularities. Math. Scand. 110, 198–222 (2012)
Pardue, K.: Deformation classes of graded modules and maximal Betti numbers. Ill. J. Math. 40, 564–585 (1996)
Spanier, E.H.: Algebraic topology. Springer, New York (1981)
Spreer, J.: A necessary condition for the tightness of odd-dimensional combinatorial manifolds. arXiv:1405.5962
Stanley, R.P.: Combinatorics and commutative algebra, 2nd edn. Birkhäuser, Boston (1996)
Swartz, E.: Thirty-five years and counting. arXiv:1411.0987
Tay, T.-S.: Lower-bound theorems for pseudomanifolds. Discrete Comput. Geom. 13, 203–216 (1995)
Terai, N., Hibi, T.: Computation of Betti numbers of monomial ideals associated with stacked polytopes. Manuscripta Math. 92, 447–453 (1997)
van Kampen, E.R.: Komplexe in euklidischen Räumen. Abh. Math. Sem. Univ. Humburg 9(72–78), 152–153 (1933)
Ziegler, G.M.: Lectures on polytopes. Graduate Texts in Mathematics, vol. 152. Springer (2007)
Acknowledgments
We would like to thank Isabella Novik for helpful comments on an earlier version of this paper. The author was partially supported by JSPS KAKENHI 25400043.
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Murai, S. Tight combinatorial manifolds and graded Betti numbers. Collect. Math. 66, 367–386 (2015). https://doi.org/10.1007/s13348-015-0137-z
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DOI: https://doi.org/10.1007/s13348-015-0137-z