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A NOTE ON BUNDLE GERBES AND INFINITE-DIMENSIONALITY

  • MICHAEL MURRAY (a1) and DANNY STEVENSON (a2)

Abstract

Let (P,Y ) be a bundle gerbe over a fibre bundle YM. We show that if M is simply connected and the fibres of YM are connected and finite-dimensional, then the Dixmier–Douady class of (P,Y ) is torsion. This corrects and extends an earlier result of the first author.

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Copyright

Corresponding author

For correspondence; e-mail: michael.murray@adelaide.edu.au

Footnotes

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The first author acknowledges the support of the Australian Research Council.

Footnotes

References

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[1]Atiyah, M. and Segal, G., ‘Twisted K-theory’, Ukr. Mat. Visn. 1 (2004), 287330.
[2]Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82 (Springer, New York–Berlin, 1982).
[3]Bouwknegt, P., Carey, A. L., Mathai, V., Murray, M. K. and Stevenson, D., ‘Twisted K-theory and K-theory of bundle gerbes’, Comm. Math. Phys. 228 (2002), 1745.
[4]Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, 107 (Birkhäuser, Boston, MA, 1993).
[5]Carey, A. L., Mickelsson, J. and Murray, M. K., ‘Bundle gerbes applied to quantum field theory’, Rev. Math. Phys. 12 (2000), 6590.
[6]Carey, A. L. and Murray, M. K., ‘Holonomy and the Wess–Zumino term’, Lett. Math. Phys. 12 (1986), 323328.
[7]Gotay, M. J., Lashof, R., Śniatycki, J. and Weinstein, A., ‘Closed forms on symplectic fibre bundles’, Comment. Math. Helv. 58 (1983), 617621.
[8]Johnson, S., ‘Constructions with bundle gerbes’, PhD Thesis, University of Adelaide, 2003, available as arXiv:math/0312175v1.
[9]Murray, M. K., ‘Bundle gerbes’, J. London Math. Soc. 54 (1996), 403416.
[10]Murray, M. K., ‘An introduction to bundle gerbes’, in: The Many Facets of Geometry, A Tribute to Nigel Hitchin (Oxford University Press, Oxford, 2010), pp. 237260.
[11]Murray, M. K. and Stevenson, D., ‘Bundle gerbes: stable isomorphism and local theory’, J. London Math. Soc. 62 (2002), 925937.
[12]Murray, M. K. and Stevenson, D., ‘The basic bundle gerbe on unitary groups’, J. Geom. Phys. 58(11) (2008), 15711590.
[13]Pressley, A. and Segal, G., Loop Groups, Oxford Mathematical Monographs (The Clarendon Press, Oxford, 1986).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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