Chern classes on differential K-theory

1 is a natural transformation of group-valued functors.

Let Ω * cl (. . . , K * ) ⊆ Ω * (. . . , K * ) denote the graded ring valued functors on smooth manifolds of smooth differential forms with coefficients in K * and its subfunctor of closed forms. We use the powers of the Bott element in K 2 in order to identify the functors Ω 0 (. . . , K * ) ∼ = Ω ev (. . . ) , Ω -1 (. . . , K * ) ∼ = Ω odd (. . . ) .

We therefore have natural transformations

where a only preserves the additive structure, while R is multiplicative. We consider the symmetric formal power series in infinitely many variables ch := i≥1 (e x i -1) ∈ Q[[x 1 , x 2 , . . . ]] .

We write ch i for the homogeneous component of degree i. Then there are polynomials

of degree i (where s i has degree i) such that

which maps the even form

The following theorem states that the Chern classes have unique lifts to the differential extensions which are, in addition, compatible with the group structures.

Theorem 1.1 1. For every i ≥ 1 there exists a unique natural transformation of setvalued functors on smooth manifolds ĉi : K0 → HZ 2i such that the following diagram commutes:

preserves the group structure.

Lifts of the Chern classes have previously been constructed in [Ber08]. The goal of the present paper is to give a much simpler, model-independent treatment. Further new, but not very deep, points of the present theorem are the assertions about uniqueness and the second statement. Our method of proof is different from [Ber08]. It is in fact a specialisation of a general principle already used in [BS] and [Bun09] for the construction of lifts of natural transformations between cohomology functors to their differential refinements.

In the next two paragraphs we connect the differential Chern classes on differential Ktheory with previous constructions of differential Chern classes in specific geometric situations.

If V := (V, h V , ∇ V ) is a hermitian vector bundle with connection over a manifold M, then we have the Cheeger-Simons classes

An even geometric family E over M (see [Bun02] for this notion) gives rise to a Bismut superconnection A(E) on an infinite-dimensional Hilbert space bundle H(E) over M. This superconnection A(E) = D(E) + ∇ H(E) + higher terms extends the family of Dirac operators D(E). If the kernel of D(E) is a vector bundle, then it has an induced metric h ker(D(E)) and connection ∇ ker(D(E)) obtained from ∇ H(E) by projection. We thus get an induced geometric bundle

and can define the class ĉCS i (H(E)) ∈ HZ 2i (M). One of the original goals of [Bun02],

which was not quite achieved there, was to extend this construction to the general case where we do not have a kernel bundle. Under the assumption that index(D(E)) ∈ K 0 (M) belongs to the i’th-step of the Atiyah-Hirzebruch filtration (i.e. vanishes after pull-back to any i -

). On the other hand, the geometric

and we have

M) also satisfies

) and thus gives a second differential refinement of the i’th Chern class of the index of D(E). But in general the class ĉi (E) differs from ĉi ([E, 0]). This can already be seen on the level of curvatures. Namely, we have

where ω [2i] denotes the degree-2i component of the form ω. In a sense, the present note gives the right answer to the problem considered in [Bun02].

Finally we discuss odd Chern classes. In topology, the odd Chern classes c odd i : K -1 → HZ i are related with the even Chern classes by suspension

.

In the smooth context the suspension isomorphism is replaced by the integration along S 1 × M → M. We have the following odd counterpart of Theorem 1.1.

Theorem 1.2 For odd i ∈ N there are unique natural transformations

commutes. The transformation in addition satisfies

Let π : W → B be a proper K-oriented map between manifolds. Then we have an Umkehr map π ! : K * (W ) → K * -n (B), where n = dim(W ) -dim(B). An integral index theorem is an assertion about the Chern classes c * (π ! (x)), or c odd * (π ! (x)) for x ∈ K * (W ), e.g. an expression of these classes in terms of the classes c * (x) or c odd * (x), respectively. A prototypical example is given in [Mad09]. The construction of differential lifts of Chern classes makes it possible to ask for geometric refinements of these kinds of results. An example of such a theorem related to the Pfaffian bundle will be discussed in a forthcoming paper.

Proof. Let K 0 ≃ Z × BU be a representative of the homotopy type of the classifying space of the functor K 0 . We choose by [BS, Prop 2.1] a sequence of manifolds (K k ) k≥0 together with maps

Let u ∈ K 0 (K 0 ) the universal class represented by the identity map K 0 → K 0 . By [BS, Prop. 2.6] we can further choose a sequence ûk ∈ K0 (K k ) such that I(û k ) = x *

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