On toric generators in the unitary and special unitary bordism rings

Lü, Zhi¹ ; Panov, Taras²

¹School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
²Department of Mathematics and Mechanics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia, Institute for Theoretical and Experimental Physics, Main Cheremushkinskaya St, Moscow 117218, Russia, Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny Lane, Moscow 127051, Russia

Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 2865-2893

Résumé

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions > 8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.

Détail

DOI : 10.2140/agt.2016.16.2865

Classification : 57R77, 14M25
Keywords: complex bordism, SU-bordism, toric manifold, characteristic numbers

Bibliographie

[1] A Borel, F Hirzebruch, Characteristic classes and homogeneous spaces, I, Amer. J. Math. 80 (1958) 458 | DOI

[2] V M Buchstaber, T E Panov, Toric topology, 204, Amer. Math. Soc. (2015) | DOI

[3] V M Buchstaber, T E Panov, N Ray, Spaces of polytopes and cobordism of quasitoric manifolds, Mosc. Math. J. 7 (2007) 219, 350

[4] V M Buchstaber, T Panov, N Ray, Toric genera, Int. Math. Res. Not. (2010) 3207 | DOI

[5] V M Buchstaber, N Ray, Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk 53 (1998) 139 | DOI

[6] V M Buchstaber, A V Ustinov, Coefficient rings of formal group laws, Mat. Sb. 206 (2015) 19 | DOI

[7] P E Conner, E E Floyd, Torsion in SU–bordism, 60, Amer. Math. Soc (1966) 74

[8] M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417 | DOI

[9] A Granville, Arithmetic properties of binomial coefficients, I : Binomial coefficients modulo prime powers, from: "Organic mathematics", CMS Conf. Proc. 20, Amer. Math. Soc. (1997) 253

[10] Z Lü, Equivariant bordism of 2–torus manifolds and unitary toric manifolds — a survey, preprint (2014)

[11] Z Lü, W Wang, Examples of quasitoric manifolds as special unitary manifolds, preprint (2015)

[12] M Masuda, D Y Suh, Classification problems of toric manifolds via topology, from: "Toric topology" (editors M Harada, Y Karshon, M Masuda, T Panov), Contemp. Math. 460, Amer. Math. Soc. (2008) 273 | DOI

[13] S P Novikov, Homotopy properties of Thom complexes, Mat. Sb. 57 (99) (1962) 407

[14] S D Oshanin, The signature of SU–manifolds, Mat. Zametki 13 (1973) 97

[15] N E Steenrod, Cohomology operations, 50, Princeton Univ. Press (1962)

[16] R E Stong, Notes on cobordism theory, 7, Princeton Univ. Press (1968)

[17] A Wilfong, Toric polynomial generators of complex cobordism, Algebr. Geom. Topol. 16 (2016) 1473 | DOI