Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AM] J.F. Adams and Z. Mahmud, Maps between classifying spaces, Inv. Math. 35 (1976), 1-41. MR 0423352 (54:11331)
  • [BBCG] A. Bahri, M. Bendersky, F.R. Cohen and S. Gitler, Decompositions of the polyhedral product functor with applications to moment-angle complexes anf related spaces, Proc. Natl. Acad. Sci. USA 106 (2009), 12241-12244. MR 2539227 (2010j:57036)
  • [BK] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Volume 304 of Lecture Notes in Mathematics, Springer Verlag (1972). MR 0365573 (51:1825)
  • [BP] V.M. Buchstaber and T.E. Panov, Torus Actions and Their Applications in Topology and Combinatorics, volume 24 of University Lecture Series, American Mathematical Society (2002). MR 1897064 (2003e:57039)
  • [DJ] M.W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), 417-451. MR 1104531 (92i:52012)
  • [DN] N. Dobrinskaja and D. Notbohm, Equivariant almost complex structures for quasi-toric manifolds, in preparation.
  • [DW] W.G. Dwyer and C.W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. Math. (2) 139 (1994), 395-442. MR 1274096 (95e:55019)
  • [K] A. Kustarev, Quasitoric manifolds with invariant almost complex structure, Preprint (2009).
  • [N1] D. Notbohm, Maps between classifying spaces, Math. Z. 207 (1991), 153-168. MR 1106820 (92b:55017)
  • [N2] D. Notbohm, Colorings of simplicial complexes and vector bundles over Davis-Januszkiewicz spaces, Math. Z. 266 (2010), no. 2, 399-405. MR 2678634
  • [NR1] D. Notbohm and N. Ray, On Davis-Januszkiewicz homotopy types. I. Formality and Rationalisation, Algebr. Geom. Topol. 5 (2005), 31-51. MR 2135544 (2006a:55016)
  • [NR2] D. Notbohm and N. Ray, On Davis-Januszkiewicz homotopy types. II. Completion and Globalisation, to appear in Algebr. Geom. Topol. MR 2683752
  • [O] R. Oliver, Higher limits via Steinberg representations, Comm. in Algebra 22 (1994), 1381-1393. MR 1261265 (95b:18007)
  • [S] G. Segal, Equivariant K-theory, Publ. Math., Inst. Hautes Ètud. Sci. 34 (1968), 129-151. MR 0234452 (38:2769)
  • [V] R.M. Vogt, Convenient categories of topological spaces for homotopy theory, Arch. Math. 22 (1971), 545-555. MR 0300277 (45:9323)
  • [W] Z. Wojtkowiak, On maps from holim$ \,F$ to $ Z$, Algebraic Topology, Barcelona 1986, Volume 1298 of Lecture Notes in Mathematics, Springer Verlag (1987). MR 928836 (89a:55034)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 55R25, 57R22, 05C15

Retrieve articles in all journals with MSC (2010): 55R25, 57R22, 05C15


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.

American Mathematical Society