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Toric polynomial generators of complex cobordism

Andrew Wilfong

Algebraic & Geometric Topology 16 (2016) 1473–1491

Although it is well known that the complex cobordism ring ΩU is isomorphic to the polynomial ring [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.

cobordism, toric variety, blow-up, fan
Mathematical Subject Classification 2010
Primary: 14M25, 57R77
Secondary: 52B20
Received: 11 September 2014
Revised: 10 April 2015
Accepted: 27 October 2015
Published: 1 July 2016
Andrew Wilfong
Department of Mathematics
Eastern Michigan University
515 Pray-Harrold
Ypsilanti, MI 48197
United States