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Exotic smooth structures on topological fiber bundles I


Authors: Sebastian Goette, Kiyoshi Igusa and Bruce Williams
Journal: Trans. Amer. Math. Soc. 366 (2014), 749-790
MSC (2010): Primary 57R22; Secondary 57R10, 57Q10
Published electronically: July 12, 2013
MathSciNet review: 3130316
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Abstract: When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and stably, this is a complete invariant. We give a more or less complete and self-contained exposition of this theory.

An important application is the computation of the Igusa-Klein higher Reidemeister torsion invariants of these exotic smooth structures. Namely, the higher torsion invariant is equal to the Poincaré dual of the image of the smooth structure class in the homology of the base. This is proved in the companion paper written by the first two authors.


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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

Bruce Williams
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: williams.4@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05857-6
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