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Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes

Author: Dietrich Notbohm
Journal: Trans. Amer. Math. Soc. 364 (2012), 3217-3239
MSC (2010): Primary 55R25, 57R22, 05C15
Published electronically: February 3, 2012
MathSciNet review: 2888243
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Abstract: For any $ (n-1)$-dimensional simplicial complex, we construct a particular $ n$-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar questions are also discussed for $ 2n$-dimensional real vector bundles with particular prescribed characteristic Pontrjagin and Euler classes. We also analyze which of these bundles admit a complex structure. It turns out that all these bundles are closely related to the tangent bundles of quasi-toric manifolds and moment angle complexes.

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Dietrich Notbohm
Affiliation: Department of Mathematics, Faculty of Sciences, Vrije Universiteit, De Boolelaan 1081a, 1081 HV Amsterdam, The Netherlands

Keywords: Davis-Januszkiewicz space, vector bundle, characteristic classes, coloring, simplicial complex, complex structure
Received by editor(s): June 25, 2009
Received by editor(s) in revised form: November 18, 2010
Published electronically: February 3, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.