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

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

2006-11-09

arxiv.org/abs/math/0611281v2

Mathematics

Differential Geometry

Construction of the topological K-theoretic direct image representatives shortened, considerations added about fibral Hodge sy

Scientific paper

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.

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