Abstract
A smooth stably complex manifold is said to be totally tangentially/normally split if its stably tangential/normal bundle is isomorphic to a sum of complex line bundles. It is proved that each class of degree greater than 2 in the graded unitary cobordism ring contains a quasitoric totally tangentially and normally split manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. M. Buchstaber, T. E. Panov, and N. Ray, “Spaces of polytopes and cobordism of quasitoric manifolds,” Mosc. Math. J. 7(2), 219–242 (2007).
N. Ray, “On a construction in bordism theory,” Proc. Edinburgh Math. Soc. (2) 29(3), 413–422 (1986).
V. M. Buchstaber and N. Ray, “Toric manifolds and complex cobordisms,” Usp. Mat. Nauk 53(2(320)), 139–140 (1998) [Uspekhi Mat. Nauk 53 (2), 371–373 (1998)].
V. M. Buchstaber and T. E. Panov, Toric Topology (Amer. Math. Soc., Providence, RI, 2015).
J. Milnor, “On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds,” Topology 3(3), 223–230 (1965).
M. Davis and T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions,” Duke Math. J. 62(2), 417–451 (1991).
A. Wilfong, “Toric polynomial generators of complex cobordism,” Algebr. Geom. Topol. 16(3), 1473–1491 (2016).
G. D. Solomadin and Yu. M. Ustinovskiy, “Projective toric polynomial generators in the unitary cobordism ring,” Mat. Sb. 207(11), 127–152 (2016) [Sb. Math. 207 (11), 1601–1624 (2016)].
S. Ochanine, “Sur les genres multiplicatifs définis par des integrales elliptiques,” Topology 26(2), 143–151 (1987).
R. E. Stong, Notes on Cobordism Theory (Princeton Univ. Press, Princeton, NJ, 1968; Mir, Moscow, 1973).
R. Thom, “Quelques propriétés globales des variétés différentiables,” Comment. Math. Helv. 28, 17–86 (1954).
B. Florian, “Lifting of morphisms to quotient presentations,” Manuscripta Math. 110(1), 33–44 (2003).
M. Imaoka, “Extendibility of negative vector bundles over the complex projective space,” Hiroshima Math. J. 36(1), 49–60 (2006).
Z. Lü and T. Panov, “On toric generators in the unitary and special unitary bordism groups,” Algebr. Geom. Topol. 16(5), 2865–2893 (2016).
I. R. Shafarevich, Basic Algebraic Geometry. 2. Schemes and Complex Manifolds (Springer-Verlag, Berlin, 2013).
N. J. Fine, “Binomial coefficients modulo a prime,” Amer. Math. Monthly 54, 589–592 (1947).
N. J. Hitchin, “Harmonic spinors,” Advances in Math. 14, 1–55 (1974).
A. Granville, “Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers,” in CMS Conf. Proc., Vol. 20: Organic Mathematics (Burnaby, BC, 1995) (Amer. Math. Soc., Providence, RI, 1997), pp. 253–276.
Acknowledgments
The author wishes to express gratitude to V. M. Buchstaber and T. E. Panov for many useful discussions. Thanks to a remark of Yu. Ustinovskiy, an inaccuracy in the construction of the blow-up along a submanifold of codimension 2 was corrected. Special thanks are due to the referee, whose critical comments have helped to improve the exposition. Being a winner of the competition “Young Mathematics of Russia,” the author expresses gratitude to its sponsors and jury.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 105, No. 5, pp. 771–791.
Rights and permissions
About this article
Cite this article
Solomadin, G.D. Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring. Math Notes 105, 763–780 (2019). https://doi.org/10.1134/S0001434619050134
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434619050134