2010/11/21 by Sebastian Goette, Goette, Sebastian, Kiyoshi Igusa +1
Mathematics · #57Q10 #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Primary 57R22 #Secondary 57R10
paper · pdf · doi:10.48550/arxiv.1011.4653
openalex publication_date 2010/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 [11], we associate a homology class in the total space of the bundle to each exotic smooth structure and we 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.