Skip to main content
SpringerLink
Log in

Quasitoric Totally Normally Split Manifolds

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS manifold for short) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex line bundles, respectively. In this paper we construct manifolds M such that any complex vector bundle over M is stably equivalent to a Whitney sum of complex line bundles. A quasitoric manifold shares this property if and only if it is a TNS manifold. We establish a new criterion for a quasitoric manifold M to be TNS via non-semidefiniteness of certain higher degree forms in the respective cohomology ring of M. In the family of quasitoric manifolds, this generalizes the theorem of J. Lannes about the signature of a simply connected stably complex TNS 4-manifold. We apply our criterion to show the flag property of the moment polytope for a nonsingular toric projective TNS manifold of complex dimension 3.

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

Similar content being viewed by others

References

  1. R. D. Arthan and S. R. Bullett, “The homology of MO(1) ∧∞ and MU(1) ∧∞,” J. Pure Appl. Algebra 26, 229–234 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. F. Atiyah, K-Theory (W.A. Benjamin, New York, 1967).

    MATH  Google Scholar 

  3. M. F. Atiyah and F. Hirzebruch, “Vector bundles and homogeneous spaces,” in Differential Geometry (Am. Math. Soc., Providence, RI, 1961), Proc. Symp. Pure Math. 3, pp. 7–38.

    Google Scholar 

  4. A. Ayzenberg and M. Masuda, “Volume polynomials and duality algebras of multi-fans,” Arnold Math. J. 2 (3), 329–381 (2016); arXiv: 1509.03008 [math.CO].

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Bogart, M. Contois, and J. Gubeladze, “Hom-polytopes,” Math. Z. 273 (3–4), 1267–1296 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. M. Buchstaber and V. D. Volodin, “Sharp upper and lower bounds for nestohedra,” Izv. Math. 75 (6), 1107–1133 (2011) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 75 (6), 17–46 (2011)].

    Article  MathSciNet  MATH  Google Scholar 

  7. V. M. Buchstaber and V. D. Volodin, “Combinatorial 2-truncated cubes and applications,” in Associahedra, Tamari Lattices and Related Structures (Birkhäuser, Basel, 2012), Prog. Math. 299, pp. 161–186.

    Chapter  Google Scholar 

  8. V. M. Buchstaber and T. E. Panov, Toric Topology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.

    Book  MATH  Google Scholar 

  9. M. W. Davis and T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions,” Duke Math. J. 62 (2), 417–451 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Huh, “Rota’s conjecture and positivity of algebraic cycles in permutohedral varieties,” PhD Thesis (Univ. Michigan, Ann Arbor, MI, 2014).

    Google Scholar 

  11. R. Morelli, “The K theory of a toric variety,” Adv. Math. 100 (2), 154–182 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Ochanine and L. Schwartz, “Une remarque sur les générateurs du cobordisme complexe,” Math. Z. 190, 543–557 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. V. Prasolov, Polynomials (Springer, Berlin, 2004), Algorithms Comput. Math. 11.

    Book  MATH  Google Scholar 

  14. A. V. Pukhlikov and A. G. Khovanskii, “Finitely additive measures of virtual polyhedra,” St. Petersburg Math. J. 4 (2), 337–356 (1993) [transl. from Algebra Anal. 4 (2), 161–185 (1992)].

    MathSciNet  Google Scholar 

  15. A. V. Pukhlikov and A. G. Khovanskii, “A Riemann–Roch theorem for integrals and sums of quasipolynomials over virtual polytopes,” St. Petersburg Math. J. 4 (4), 789–812 (1993) [transl. from Algebra Anal. 4 (4), 188–216 (1992)].

    MathSciNet  MATH  Google Scholar 

  16. N. Ray, “On a construction in bordism theory,” Proc. Edinb. Math. Soc. 29 (3), 413–422 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Sankaran and V. Uma, “K-theory of quasi-toric manifolds,” Osaka J. Math. 44 (1), 71–89 (2007).

    MathSciNet  MATH  Google Scholar 

  18. G. D. Solomadin, “Quasitoric totally normally split representatives in unitary cobordism ring,” Mat. Zametki, doi: 10.4213/mzm11818 (2019); arXiv: 1704.07403 [math.AT].

    Google Scholar 

  19. A. Tarski, A Decision Method for Elementary Algebra and Geometry (Rand Corp., Santa Monica, CA, 1948).

    MATH  Google Scholar 

  20. V. A. Timorin, “An analogue of the Hodge–Riemann relations for simple convex polytopes,” Russ. Math. Surv. 54 (2), 381–426 (1999) [transl. from Usp. Mat. Nauk 54 (2), 113–162 (1999)].

    Article  MATH  Google Scholar 

  21. G. M. Ziegler, Lectures on Polytopes (Springer, New York, 1995), Grad. Texts Math. 152.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Grigory D. Solomadin

Authors
  1. Grigory D. Solomadin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Grigory D. Solomadin.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Vol. 302, pp. 377–399.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Solomadin, G.D. Quasitoric Totally Normally Split Manifolds. Proc. Steklov Inst. Math. 302, 358–379 (2018). https://doi.org/10.1134/S0081543818060196

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543818060196

Search

Navigation