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

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

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 …

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

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 ima…

A Construction of String 2-Group Models using a Transgression-Regression Technique

In this note we present a new construction of the string group that ends optionally in two different contexts: strict diffeological 2-groups or finite-dimensional Lie 2-groups. It is canonical in the sense that no choices are involved; all the data is written down and can be looked up (at least somewhere). The basis of our construction is the basic gerbe of Gawedzki-Reis and Meinrenken. The main new insight is that under a transgression-regression procedure, the basic gerbe picks up a multiplicative structure coming from the Mickelsson product over the loop group. The conclusion of the constru…

端周期Dirac算子族的指标公式

An index formula for families of end-periodic Dirac operators

本文为端周期流形上的Dirac算子族建立了指标公式，通过转降公式和新的端周期eta形式，推广了已有的eta不变量。

Large time limit and local L^2-index theorems for families

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. …

Zeta Forms and the Local Family Index Theorem

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formu…