Open AccessExotic smooth structures on topological fiber bundles II
Author(s) -
Sebastian Goette,
Kiyoshi Igusa
Publication year - 2013
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/s0002-9947-2013-05858-8
Subject(s) - mathematics , pure mathematics , vector bundle , torsion (gastropod) , cohomology , homology (biology) , fiber bundle , characteristic class , topology (electrical circuits) , bundle , mathematical analysis , combinatorics , medicine , biochemistry , chemistry , materials science , surgery , composite material , gene
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.