Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 1

2006/11/09 by Alain Berthomieu, Berthomieu, Alain
Mathematics · #14F05 #19E20 #57R20 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #math.DG #math.KT #msc:14F05 #msc:19E20 #msc:57R20

paper · pdf · doi:10.48550/arxiv.math/0611281

Construction of the topological K-theoretic direct image representatives shortened, considerations added about fibral Hodge symmetry and name of ``transgressive K-theory'' changed to ''free multiplicative K-theory''

openalex publication_date 2006/11/09 · arxiv created 2007/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper contains the constructions of a real manifold version of relative K-theory, and of an extension of Karoubi's multiplicative K-theory suggested by U. Bunke (which I call ``free multiplicative K-theory'' in the sequel). Chern-Simons-Nadel type classes on relative K-theory are constructed, while it is proved that on free multiplicative K-theory, there is a notion of Chern-Weil character form, and of a Borel-type characteristic class (which is a differential form modulo exact forms) which recovers the classes c_k of flat vector bundles studied by Bismut and Lott. Finally, a direct image for relative K-theory under proper submersion of compact orientable real manifolds is constructed.

Related

• Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 2
• Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 3
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