Toric polynomial generators of complex cobordism

Wilfong, Andrew

doi:10.2140/agt.2016.16.1473

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Toric polynomial generators of complex cobordism

Andrew Wilfong

Algebraic & Geometric Topology 16 (2016) 1473–1491

DOI: 10.2140/agt.2016.16.1473

Abstract

Although it is well known that the complex cobordism ring $Ω_{*}^{U}$ is isomorphic to the polynomial ring $ℤ [α_{1}, α_{2}, \dots]$ , an explicit description for convenient generators $α_{1}, α_{2}, \dots$ 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.

Keywords

cobordism, toric variety, blow-up, fan

Mathematical Subject Classification 2010

Primary: 14M25, 57R77

Secondary: 52B20

References

Publication

Received: 11 September 2014

Revised: 10 April 2015

Accepted: 27 October 2015

Published: 1 July 2016

Authors

Andrew Wilfong
Department of Mathematics Eastern Michigan University 515 Pray-Harrold Ypsilanti, MI 48197 United States

Volume 16, issue 3 (2016)