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Exotic smooth structures on topological fiber bundles II


Authors: Sebastian Goette and Kiyoshi Igusa
Journal: Trans. Amer. Math. Soc. 366 (2014), 791-832
MSC (2010): Primary 57R22; Secondary 57R10, 57Q10
DOI: https://doi.org/10.1090/S0002-9947-2013-05858-8
Published electronically: July 12, 2013
MathSciNet review: 3130317
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Abstract: We use a variation of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension plus 3). Using a variation of the Dwyer-Weiss-Williams smoothing theory which we explain in a separate joint paper with Bruce Williams, we associate a homology class in the total space of the bundle to each exotic smooth structure and show that the image of this class in the homology of the base is the Poincaré dual of the relative higher Igusa-Klein (IK) torsion invariant. This answers the question, in the relative case, of which cohomology classes can occur as relative higher torsion classes.


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Additional Information

Sebastian Goette
Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
Email: sebastian.goette@math.uni-freiburg.de

Kiyoshi Igusa
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454
Email: igusa@brandeis.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05858-8
Received by editor(s): December 5, 2010
Received by editor(s) in revised form: March 17, 2012, and April 7, 2012
Published electronically: July 12, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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