Abstract
Let M be a manifold that is given by the intersection of k quadrics in \(\mathbb {R}^{n}\) with the unit sphere, such that it is symmetric with respect to the n coordinate hyperplanes. Let P be the quotient of this manifold by the action of \(\mathbb {Z}_{2}^{n}\) (as group of reflections). P is a simple polytope and M is determined by P, so the homology groups of M are determined by the combinatorial structure of P. And P is associated to an oriented matroid. In this work, we explore the relation between the topes of this oriented matroid and the topology of M. Then we consider the case in which P is the dual polytope of a cyclic polytope Q. When \(k=3\), we prove that M is a connected sum of sphere products, including the four-dimensional case. Finally, we calculate the Betti numbers of M by means of a cell decomposition of the plane associated to the oriented matroid, looking for new bridges between different areas of the mathematics.
Similar content being viewed by others
References
Bosio, F.: Un invariant remarquable des polytopes simples (2011). arXiv:1110.0399
Bosio, F.: Diffeomorphic moment-angle manifolds with different Betti numbers (2014). arXiv:1410.3304
Bosio, F., Meersseman, L.: Real quadrics in \({\mathbb{C}}^n\), complex manifolds and convex polytopes. Acta Math. 197(1), 53–127 (2006)
Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of mathematics and its applications, vol. 46. Cambridge University Press, Cambridge (1993)
Cordovil, R., Duchet, P.: Cyclic polytopes and oriented matroids. Eur. J. Comb. 21, 49–64 (2000)
De Loera, J., Rambau, J., Santos, F.: Triangulations: Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, Heidelberg (2010)
Gale, D.: Neighborly and cyclic polytopes. Proc. Symp. Pure Math. 7(Convexity), 225–232 (1963)
Gitler, S., López de Medrano, S.: Intersections of quadrics, moment-angle manifolds and connected sums. Geom. Topol. 17, 1497–1534 (2013). http://www.msp.warwick.ac.uk/gt/2013/17-03/p034.xhtml. doi:10.2140/gt.2013.17.1497
Gómez-Gutiérrez, V., López de Medrano, S.: Topology of the intersections of quadrics II. Bol. Soc. Math. Mex. 20, 237–255 (2014)
Grünbaum, B.: Convex Polytopes, 2nd edn. Springer-Verlag, New York (2003)
Kim, E.D., Santos, F.: Companion to an update to the Hirsch conjecture. arXiv:0912.4235
Kim, E.D., Santos, F.: An update on the Hirsch conjecture. Jahresbericht der Deustchen Mathematiker Vereinigung 112(2), 73–98 (2010)
López de Medrano, S.: The Topology of the Intersection of Quadrics in \({\mathbb{R}}^{n}\), In: Algebraic Topology (Arcata Ca,1986), Lecture Notes in Mathematics, vol. 1370, pp. 280–292. Springer-Verlag, Berlin (1989)
Migliore, J., Nagel, U.: Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers. Adv. Math. 180, 1–63 (2003)
Padrol, A.: Neighborly and almost neighborly configurations, and their duals. Doctoral Thesis, Universitat Politèctnica de Catalunya (2013)
Sturmfels, B.: Neighborly polytopes and oriented matroids. Eur. J. Comb. 9, 537–546 (1988)
Sturmfels, B.: Some applications of affine gale diagrams to polytopes with few vertices. SIAM J. Discrete Math. 1(1), 121–133 (1988)
Schenzel, P.: Über die freien Auflösungen extremaler Cohen-Macaulay Ringe. J. Algebra 64, 93–101 (1980)
Terai, N., Hibi, T.: Computation of Betti numbers of monomial ideals associated with cyclic polytopes. Discrete Comput. Geom. 15, 287–295 (1996)
Wall, C.T.C.: Stability, pencils and polytopes. Bull. Lond. Math. Soc. 12, 401–421 (1980)
Ziegler, G.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)
Acknowledgments
The author was partially supported by project PAPIIT-DGAPA IN111415.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Samuel Gitler, in memoriam.
Rights and permissions
About this article
Cite this article
Gómez-Gutiérrez, V. Cyclic polytopes, oriented matroids and intersections of quadrics. Bol. Soc. Mat. Mex. 23, 87–118 (2017). https://doi.org/10.1007/s40590-016-0144-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40590-016-0144-4
Keywords
- Cyclic polytopes
- Oriented matroids
- Moment-angle manifolds
- Intersections of quadrics
- Arrangement of lines