Geodesic
Toric polynomial generators of complex cobordism
Wilfong, Andrew1
1Department of Mathematics, Eastern Michigan University, 515 Pray-Harrold, Ypsilanti, MI 48197, United States
Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1473-1491
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Although it is well known that the complex cobordism ring Ω∗U is isomorphic to the polynomial ring ℤ[α1,α2,…], an explicit description for convenient generators α1,α2,… has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well.

DOI : 10.2140/agt.2016.16.1473
Classification : 14M25, 57R77, 52B20
Keywords: cobordism, toric variety, blow-up, fan

Wilfong, Andrew  1

1 Department of Mathematics, Eastern Michigan University, 515 Pray-Harrold, Ypsilanti, MI 48197, United States 
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Wilfong, Andrew. Toric polynomial generators of complex cobordism. Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1473-1491. doi: 10.2140/agt.2016.16.1473

[1] V V Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. 43 (1991) 569

[2] V M Buchstaber, T E Panov, Torus actions and their applications in topology and combinatorics, 24, Amer. Math. Soc. (2002)

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

[4] V M Buchstaber, N Ray, Tangential structures on toric manifolds, and connected sums of polytopes, Internat. Math. Res. Notices (2001) 193

[5] D A Cox, J B Little, H K Schenck, Toric varieties, 124, Amer. Math. Soc. (2011)

[6] V I Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978) 85, 247

[7] A Dickenstein, S Di Rocco, R Piene, Classifying smooth lattice polytopes via toric fibrations, Adv. Math. 222 (2009) 240

[8] W Fulton, Introduction to toric varieties, 131, Princeton University (1993)

[9] P Griffiths, J Harris, Principles of algebraic geometry, Wiley (1978)

[10] D Huybrechts, Complex geometry: an introduction, Springer (2005)

[11] B Johnston, The values of the Milnor genus on smooth projective connected complex varieties, Topology Appl. 138 (2004) 189

[12] J Jurkiewicz, Torus embeddings, polyhedra, k∗–actions and homology, Dissertationes Math. (Rozprawy Mat.) 236 (1985)

[13] P Kleinschmidt, A classification of toric varieties with few generators, Aequationes Math. 35 (1988) 254

[14] S P Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk SSSR 132 (1960) 1031

[15] R E Stong, Notes on cobordism theory, Princeton University; University of Tokyo (1968)

[16] R Thom, Travaux de Milnor sur le cobordisme, from: "Séminaire Bourbaki, 1958∕59 (Exposé 180)", W A Benjamin (1965)

[17] A Wilfong, Toric varieties and cobordism, PhD thesis, University of Kentucky (2013)

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