DIRECT IMAGE FOR SOME SECONDARY K-THEORIES

The real counterpart of relative K-theory (considered in the complex setting in [4]) is considered here, some direct image under proper submersion is constructed, and a Grothendieck-Riemann-Roch theorem for Johnson-Nadel-Chern-Simons classes is proved. Metric properties are also studied. This needs to revisit the construction of eta-forms in the case where the direct image is provided by the vertical Euler (de Rham) operator. A direct image under proper submersions of some "non hermitian smooth" or "free multiplicative" K-theory is deduced (in the same context). Double submersions are also studied to establish some functoriality properties of these direct images.