Smooth K-Theory

Nous construisons une version differentiable du charactère de Chern ĉh : K(B) → Ĥ(B, Q), où Ĥ(B, Q) est une extension differentiable de la cohomologie rationelle, et nous demontrons que ĉh induit un isomorphisme rationel.

Si p : W → B est une submersion propre avec une K-orientation differentiable, nous definissons une classe A(p) ∈ Ĥev (W, Q) (compare Lemma 6.17 -In this paper we construct a model of a smooth extension of the generalized cohomology theory K, complex K-theory. Historically, the concept of smooth extensions of a cohomology theory started with smooth integral cohomology [CS85], also called real Deligne cohomology, see [Bry93]. A second, geometric model of smooth integral cohomology is given in [CS85], where the smooth integral cohomology classes were called differential characters. One important motivation of its definition was that one can associate natural differential characters to hermitean vector bundles with connection which refine the Chern classes. The differential character in degree two even classifies hermitean line bundles with connection up to isomorphism. The multiplicative structure of smooth integral cohomology also encodes cohomology operations, see [Gom].

The holomorphic counterpart of the theory became an important ingredient of arithmetic geometry. 1.1.2. -Motivated by the problem of setting up lagrangians for quantum field theories with differential form field strength it was argued in [FH00], [Fre00] that one may need smooth extensions of other generalized cohomology theories. The choice of the generalized cohomology theory is here dictated by a charge quantization condition, which mathematically is reflected by a lattice in real cohomology. Let N be a graded real vector space such that the field strength lives in Ω d=0 (B) ⊗ N , the closed forms on the manifold B with coefficients in N . Let L(B) ⊂ H(B, N ) be the lattice given by the charge quantization condition on B. Then one looks for a generalized cohomology theory h and a natural transformation c : h(B) → H(B, N ) such that c(h(B)) = L(B). It was argued in [FH00], [Fre00] that the fields of the theory should be considered as cycles for a smooth extension ĥ of the pair (h, c). For example, if N = R and the charge quantization leads to L(B) = im(H(B, Z) → H(B, R)), then the relevant smooth extension could be the smooth integral cohomology theory of [CS85].

In Subsection 1.2 we will introduce the notion of a smooth extension in an axiomatic way. 1.1.3. - [Fre00] proposes in particular to consider smooth extensions of complex and real versions of K-theory. In that paper it was furthermore indicated how cycle models of such smooth extensions could look like. The goal of the present paper is to carry through this program in the case of complex K-theory. 1.1.4. -In the remainder of the present subsection we describe, expanding the abstract, our main results. The main ingredient is a construction of an analytic model of smooth K-theory (1) using cycles and relations. 1.1.5. -Our philosophy for the construction of smooth K-theory is that a vector bundle with connection or a family of Dirac operators with some additional geometry should represent a smooth K-theory class tautologically. In this way we follow the outline in [Fre00]. Our class of cycles is quite big. This makes the construction of smooth K-theory classes or transformations to smooth K-theory easy, but it complicates the verification that certain cycle level constructions out of smooth Ktheory are well-defined. The great advantage of our choice is that the constructions of the product and the push-forward on the level of cycles are of differential geometric nature.

More precisely we use the notion of a geometric family which was introduced in [Bun] in order to subsume all geometric data needed to define a Bismut superconnection in one notion. A cycle of the smooth K-theory K(B) of a compact manifold B is a pair (E, ρ) of a geometric family E and an element ρ ∈ Ω(B)/im(d), see Section 2. Therefore, cycles are differential geometric objects. Secondary spectral invariants from local index theory, namely η-forms, enter the definition of the relations (see Definition 2.10). The first main result is that our construction really yields a smooth extension in the sense of Definition 1.1. 1.1.6. -Our smooth K-theory K(B) is a contravariant functor on the category of compact smooth manifolds (possibly with boundary) with values in the category of Z/2

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