Skip to main content
SpringerLink
Log in

Samuel Gitler and the topology of intersections of quadrics

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Notes

  1. I had also a very friendly relationship with Steenrod, but this had naturally and gradually grown starting with his course in Mexico.

  2. This section includes several technicalities, but readers not familiar with Topology need not stop reading, just consider them as labels (complex, real, (A), (a)...) that can be of some help in following in the story.

  3. I found the first example of such a connected sum in the work of Dennis McGavran [35] on torus actions, after a long search in Mathematical Reviews, Google not being available at the time. This was a big surprise and an extremely valuable heuristic guide for our work. On the other hand, only one of McGavran’s examples was relevant for my problem. Furthermore, all his arguments are valid only in the PL-category and I have not been able to reconstruct his proofs in the smooth case, so the proof of my results had to be totally different from his. In the real case (see below) one obtains easily the surface of genus 5, so the fact that such connected sums appear in this question becomes less surprising.

  4. This is a simple illustration of the fact that many topological properties of \(M_{1}\) and its generalizations are related to convexity properties of the coefficients.

  5. This time it was Pepe who invited me, but clearly that was the result of a plot that included Alberto and Xavier Gómez Mont.

  6. Up to now we do not know explicit diffeomorphisms between our varieties and the connected sums, only their existence by the h-cobordism theorem.

  7. This took place during a conference for my 65th birthday, organized essentially by Pepe. Actually, Pepe organized everything from abroad through the web and did not even come to the conference, the local organizer was my former student León Kushner.

  8. Actually, Laurent had told me that a referee suggested to them to include in the published version of [12] a connection with another work on the same objects, but this never attracted my attention. In the same way, had I accepted a conference for my 60th anniversary, Sam’s talk would have been different and maybe none of this would have happened.

  9. Recall the beginning of this section.

  10. Some readers must know what this meant: after three or four hours where Sam had spoken about several different questions (including ideas that he had long ago or had just occurred to him that day during breakfast) one is totally exhausted, while he is ready to go on for several more hours.

  11. This lattice is equivalent to the lattice of points in \(\mathbf {R}^m\) with integral non-negative coordinates, where m is the number of facets of the polytope P corresponding to Z. In terms of polytopes this is given by the map \(J\mapsto P(J)\) (see [6], Theorem 2.4).

  12. These examples are still quite geometric and simple. It should be said that the topologically more complicated examples, in particular those with non-trivial Massey products, are also numerous. See [18].

References

  1. Agrachev, A.A.: Topology of quadratic maps and Hessians of smooth maps. Itogi nauki. VINITI. Algebra Topol. Geom. 26, 85–124 (1988)

    Google Scholar 

  2. Al-Raisi, A.: Equivariance, Module Structure, Branched Covers, Strickland Maps and Cohomology related to the Polyhedral Product Functor. Ph. D. Thesis, University of Rochester (2014)

  3. Artal, E., López de Medrano, S., Lozano, M.T.: The dodecahedron: from intersections of quadrics to Borromean rings. In: López, M.C., Martín Peinador, E., Rodríguez Sanjurjo, J.M., Ruiz Sancho, J.M. (eds.) A Mathematical Tribute to Professor José María Montesinos Amilibia. Universidad Complutense de Madrid, pp. 85–103 (2016). ISBN: 978-84-608-1684-3

  4. Ayzenberg, A., Buchstaber, V.: Nerve complexes and moment-angle spaces of convex polytopes. Proc. Steklov Inst. Math. 275, 15–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces. Adv. Math. 22, 1634–1668 (2010)

    Article  MATH  Google Scholar 

  6. Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: Operations on polyhedral products and a new topological construction of infinite families of toric manifolds (2015). arXiv:1011.0094v5

  7. Barreto, Y., López de Medrano, S., Verjovsky, A.: Some Open Book and Contact Structures on Moment-Angle Manifolds. arXiv:1510.07729. Boletín de la Sociedad Matemática Mexicana, this volume

  8. Barreto, Y., Verjovsky, A.: Moment-angle manifolds, intersection of quadrics and higher dimensional contact manifolds. Mosc. Math. J. 14(4), 669–696 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Borman, M.S., Eliashberg, Y., Murphy, E.: Existence and classification of overtwisted contact structures in all dimensions. Acta Math. 215, 281–361 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bosio, F.: Un invariant remarquable des polytopes simples (2011). arXiv:1110.0399

  11. Bosio, F.: Diffeomorphic moment-angle manifolds with different Betti numbers (2014). arXiv:1410.3304

  12. Bosio, F., Meersseman, L.: Real quadrics in \({\varvec {C}^{n}}\), complex manifolds and convex polytopes. Acta Math. 197, 53–127 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Buchstaber, V., Panov, T.: Torus Actions and Their Applications in Topology and Combinatorics. University Lecture Series. AMS, vol. 24, New York (2002)

  14. Buchstaber, V., Panov, T.: Toric Topology (2014). arXiv:1210.2368

  15. Cai, L.: Cup and cap products in real moment-angle manifolds. Ph.D. Thesis, University of Kyushu (2014)

  16. Davis, M., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62, 417–451 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Degtyarev, A., Itenberg, I., Kharlamov, V.: On the number of components of a complete intersection of real quadrics. In: Itenberg, I., Jöricke, B., Passare, M. (eds.) Perspectives in Analysis, Geometry, and Topology (On the Occasion of the 60th Birthday of Oleg Viro). Progress in Mathematics, vol. 296, pp. 81–107 (2012)

  18. Denham, G., Suciu, A.: Moment angle complexes, monomial ideals, and Massey products. Pure Appl. Math. Q. 3(1), 25–60 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Franz, M., López de Medrano S., Malik, J.: Mutants of compactified representations revisited. arXiv:1511.01115. Boletín de la Sociedad Matemática Mexicana, this volume

  20. Gitler, S., López de Medrano, S.: Intersections of quadrics, moment-angle manifolds and connected sums. Geom. Topol. 17, 1497–1534 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gómez Gutiérrez, V.: Cyclic polytopes and intersections of quadrics. Boletín de la Sociedad Matemática Mexicana, this volume

  22. Gómez Gutiérrez, V., López de Medrano, S.: An extension of the Toeplitz–Hausdorff theorem. Boletín de la Sociedad Matemática Mexicana 9(2), 273–278 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Gómez Gutiérrez, V., López de Medrano, S.: Stably parallelizable compact manifolds are complete intersections of quadrics. Instituto de Matemáticas, UNAM (Preprint)

  24. Gómez Gutiérrez, V., López de Medrano, S.: Surfaces as complete intersections, in Riemann and Klein surfaces, automorphisms, symmetries and moduli spaces. Contemp. Math. 629, 171–180 (2014)

  25. Gómez Gutiérrez, V., López de Medrano, S.: Topology of the intersection of quadrics II. Bol. Soc. Mat. Mex. 20, 237–255 (2014)

  26. Grunbaum, B.: Convex Polytopes. Graduate Texts in Mathematics, vol. 221. Springer-Verlag, New York Berlin Heidelberg (2003)

  27. Katzarkov, L., Lupercio, E., Meersseman, L., Verjovsky, A.: The definition of a non-commutative toric variety. In: Tillmann, U., Galatius, S., Sinha, D. (eds.) Algebraic Topology: Applications and New Directions. Contemporary Mathematics, vol. 620. AMS pp. 223–250 (2014)

  28. Krasnov, V.A.: Cohomology of real three-dimensional triquadrics. Izv. Math. 76, 113–138 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lerario, A.: Convex pencils of real quadratic forms. Discrete Comput. Geom. 48, 1025–1047 (2012)

    MathSciNet  MATH  Google Scholar 

  30. López de Medrano, S.: The space of Siegel leaves of a holomorphic vector field. In: Gómez-Mont, X., Seade, J.A., Verjovsky, A. (eds.) Holomorphic Dynamics (Mexico, 1986). Lecture Notes in Mathematics, vol. 1345, pp. 233–245. Springer, Berlin (1988)

  31. López de Medrano, S.: The Topology of the Intersection of Quadrics in \({\varvec {R}^{n}}\). In: Carlsson, G., Cohen, R., Miller, H.R., Ravenel, D.C. (eds.) Algebraic Topology (Arcata Ca, 1986). Lecture Notes in Mathematics, vol. 1370, pp. 280–292. Springer, Berlin, Heidelberg (1989)

  32. López de Medrano, S.: Singularities of real homogeneous quadratic mappings. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, vol. 108, pp. 95–112. Springer-Verlag (2014)

  33. López de Medrano, S.: Singular Intersections of Quadrics I (to appear in Trends in Mathematics, special volume dedicated to José Seade’s 60th birthday)

  34. López de Medrano, S., Verjovsky, A.: A new family of complex, compact, non-symplectic manifolds. Bol. Soc. Brasil. Math. 28, 23–269 (1997)

    MathSciNet  MATH  Google Scholar 

  35. McGavran, D.: Adjacent connected sums and torus actions. Trans. Am. Math. Soc. 251, 235–254 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. Meersseman, L.: A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317, 79–115 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Padrol, A.: Neighborly and almost neighborly configurations, and their duals. Doctoral Thesis, Universitat Politèctnica de Catalunya (2013)

  38. Wall, C.T.C.: Stability, pencils and polytopes. Bull. Lond. Math. Soc. 12, 401–421 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F., 04510, Mexico

    Santiago López de Medrano

Authors
  1. Santiago López de Medrano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Santiago López de Medrano.

Additional information

In memory of Sam Gitler.

Partially supported by Project PAPIIT-DGAPA IN111415.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

López de Medrano, S. Samuel Gitler and the topology of intersections of quadrics. Bol. Soc. Mat. Mex. 23, 5–21 (2017). https://doi.org/10.1007/s40590-016-0148-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40590-016-0148-0

Mathematics Subject Classification

Search

Navigation