## Tautological classes and smooth bundles over BSU(2)

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## Abstract:

For a Lie group $G$ and a smooth manifold $W$, we study the difference between smooth actions of $G$ on $W$ and fiber bundles over the classifying space of $G$ with fiber $W$ and structure group $\mathrm {Diff}(W)$. In particular, we exhibit smooth manifold bundles over $B\mathrm {SU}(2)$ that are not induced by an action. The main tool for reaching this goal is a technical result that gives a constraint for the values of tautological classes of the fiber bundle associated to a group action.## References

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## Additional Information

**Jens Reinhold**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- ORCID: 0000-0002-7445-8204
- Email: jreinh@stanford.edu
- Received by editor(s): February 26, 2018
- Received by editor(s) in revised form: May 17, 2018
- Published electronically: November 13, 2018
- Additional Notes: The author was supported by the E. K. Potter Stanford Graduate Fellowship. A substantial part of the research outlined in this work was done during two very pleasurable visits at the Department of Mathematics at the University of Copenhagen, which were financially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 682922).
- Communicated by: Mark Behrens
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 885-895 - MSC (2010): Primary 55R37, 57S05, 57S25
- DOI: https://doi.org/10.1090/proc/14249
- MathSciNet review: 3894925