Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes

Notbohm, Dietrich

doi:10.1090/S0002-9947-2012-05508-5

Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes
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by Dietrich Notbohm

Trans. Amer. Math. Soc. 364 (2012), 3217-3239

DOI: https://doi.org/10.1090/S0002-9947-2012-05508-5

Published electronically: February 3, 2012

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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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