Vector bundles over DavisJanuszkiewicz spaces with prescribed characteristic classes
Author:
Dietrich Notbohm
Journal:
Trans. Amer. Math. Soc. 364 (2012), 32173239
MSC (2010):
Primary 55R25, 57R22, 05C15
Published electronically:
February 3, 2012
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Abstract: For any dimensional simplicial complex, we construct a particular dimensional complex vector bundle over the associated DavisJanuszkiewicz 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 quasitoric manifolds and moment angle complexes.
 [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 momentangle complexes and related spaces, Proc. Natl.
Acad. Sci. USA 106 (2009), no. 30, 12241–12244.
MR
2539227 (2010j:57036), http://dx.doi.org/10.1073/pnas.0905159106
 [BK]
A.
K. Bousfield and D.
M. Kan, Homotopy limits, completions and localizations,
Lecture Notes in Mathematics, Vol. 304, SpringerVerlag, BerlinNew York,
1972. MR
0365573 (51 #1825)
 [BP]
Victor
M. Buchstaber and Taras
E. Panov, Torus actions and their applications in topology and
combinatorics, University Lecture Series, vol. 24, American
Mathematical Society, Providence, RI, 2002. MR 1897064
(2003e:57039)
 [DJ]
Michael
W. Davis and Tadeusz
Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus
actions, Duke Math. J. 62 (1991), no. 2,
417–451. MR 1104531
(92i:52012), http://dx.doi.org/10.1215/S0012709491062174
 [DN]
N. Dobrinskaja and D. Notbohm, Equivariant almost complex structures for quasitoric manifolds, in preparation.
 [DW]
W.
G. Dwyer and C.
W. Wilkerson, Homotopy fixedpoint methods for Lie groups and
finite loop spaces, Ann. of Math. (2) 139 (1994),
no. 2, 395–442. MR 1274096
(95e:55019), http://dx.doi.org/10.2307/2946585
 [K]
A. Kustarev, Quasitoric manifolds with invariant almost complex structure, Preprint (2009).
 [N1]
Dietrich
Notbohm, Maps between classifying spaces, Math. Z.
207 (1991), no. 1, 153–168. MR 1106820
(92b:55017), http://dx.doi.org/10.1007/BF02571382
 [N2]
Dietrich
Notbohm, Colorings of simplicial complexes and vector bundles over
DavisJanuszkiewicz spaces, Math. Z. 266 (2010),
no. 2, 399–405. MR 2678634
(2011i:55020), http://dx.doi.org/10.1007/s002090090575y
 [NR1]
Dietrich
Notbohm and Nigel
Ray, On DavisJanuszkiewicz homotopy types. I. Formality and
rationalisation, Algebr. Geom. Topol. 5 (2005),
31–51 (electronic). MR 2135544
(2006a:55016), http://dx.doi.org/10.2140/agt.2005.5.31
 [NR2]
Dietrich
Notbohm and Nigel
Ray, On DavisJanuszkiewicz homotopy types II: completion and
globalisation, Algebr. Geom. Topol. 10 (2010),
no. 3, 1747–1780. MR 2683752
(2011j:55015), http://dx.doi.org/10.2140/agt.2010.10.1747
 [O]
Bob
Oliver, Higher limits via Steinberg representations, Comm.
Algebra 22 (1994), no. 4, 1381–1393. MR 1261265
(95b:18007), http://dx.doi.org/10.1080/00927879408824911
 [S]
Graeme
Segal, Equivariant 𝐾theory, Inst. Hautes
Études Sci. Publ. Math. 34 (1968), 129–151.
MR
0234452 (38 #2769)
 [V]
Rainer
M. Vogt, Convenient categories of topological spaces for homotopy
theory, Arch. Math. (Basel) 22 (1971), 545–555.
MR
0300277 (45 #9323)
 [W]
Zdzisław
Wojtkowiak, On maps from ℎ𝑜\varinjlim𝐹 to
𝑍, Algebraic topology, Barcelona, 1986, Lecture Notes in
Math., vol. 1298, Springer, Berlin, 1987, pp. 227–236. MR 928836
(89a:55034), http://dx.doi.org/10.1007/BFb0083013
 [AM]
 J.F. Adams and Z. Mahmud, Maps between classifying spaces, Inv. Math. 35 (1976), 141. MR 0423352 (54:11331)
 [BBCG]
 A. Bahri, M. Bendersky, F.R. Cohen and S. Gitler, Decompositions of the polyhedral product functor with applications to momentangle complexes anf related spaces, Proc. Natl. Acad. Sci. USA 106 (2009), 1224112244. 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), 417451. MR 1104531 (92i:52012)
 [DN]
 N. Dobrinskaja and D. Notbohm, Equivariant almost complex structures for quasitoric manifolds, in preparation.
 [DW]
 W.G. Dwyer and C.W. Wilkerson, Homotopy fixedpoint methods for Lie groups and finite loop spaces, Ann. Math. (2) 139 (1994), 395442. 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), 153168. MR 1106820 (92b:55017)
 [N2]
 D. Notbohm, Colorings of simplicial complexes and vector bundles over DavisJanuszkiewicz spaces, Math. Z. 266 (2010), no. 2, 399405. MR 2678634
 [NR1]
 D. Notbohm and N. Ray, On DavisJanuszkiewicz homotopy types. I. Formality and Rationalisation, Algebr. Geom. Topol. 5 (2005), 3151. MR 2135544 (2006a:55016)
 [NR2]
 D. Notbohm and N. Ray, On DavisJanuszkiewicz 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), 13811393. MR 1261265 (95b:18007)
 [S]
 G. Segal, Equivariant Ktheory, Publ. Math., Inst. Hautes Ètud. Sci. 34 (1968), 129151. MR 0234452 (38:2769)
 [V]
 R.M. Vogt, Convenient categories of topological spaces for homotopy theory, Arch. Math. 22 (1971), 545555. MR 0300277 (45:9323)
 [W]
 Z. Wojtkowiak, On maps from holim to , Algebraic Topology, Barcelona 1986, Volume 1298 of Lecture Notes in Mathematics, Springer Verlag (1987). MR 928836 (89a:55034)
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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:
http://dx.doi.org/10.1090/S000299472012055085
PII:
S 00029947(2012)055085
Keywords:
DavisJanuszkiewicz 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.
