2007/03/30 by Alain Berthomieu, Berthomieu, Alain
Mathematics · #57R20 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.DG #msc:57R20
paper · pdf · doi:10.48550/arxiv.math/0703916
arxiv created 2007/03/30 · openalex publication_date 2007/03/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is the sequel of the first part math.DG/0611281. Here, the procedure of transgressing the families index theorem (the so-called η-form) is adapted to take in account the case of Dirac type operators with kernels of varying dimension. The constructed form is then used to define the direct image under proper submersions of the ``free multiplicative'' K-theory which was defined in the first part, the behaviour of the characteristic classes on free multiplicative K-theory under submersion is studied. Some universal caracterisation of the forms is provided. Finally, combining our result with Bismut and Lott's results on direct images of flat vector bundles yields a ``Grothendieck-Riemann-Roch'' theorem for Nadel-Chern-Simons classes on relative K-theory for flat vector bundles which was defined in the first part.