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

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

Published electronically:
February 3, 2012

MathSciNet review:
2888243

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Abstract: For any -dimensional simplicial complex, we construct a particular -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 -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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Additional Information

**Dietrich Notbohm**

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

Email:
notbohm@few.vu.nl

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

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.