ar X iv : 0 70 7 . 00 46 v 2 [ m at h . K T ] 2 3 M ay 2 00 8 SMOOTH K - THEORY by Ulrich Bunke & Thomas Schick
نویسنده
چکیده
— In this paper we consider smooth extensions of cohomology theories. In particular we construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion p : W → B and define the associated push-forward p̂! : K̂(W ) → K̂(B). We show that the push-forward has the expected properties as functoriality, compatibility with pull-back diagrams, projection formula and a bordism formula. We construct a multiplicative lift of the Chern character ĉh : K̂(B) → Ĥ(B,Q), where Ĥ(B,Q) denotes the smooth extension of rational cohomology, and we show that ĉh induces a rational isomorphism If p : W → B is a proper submersion with a smooth K-orientation, then we define a class A(p) ∈ Ĥ(W,Q) and the modified push-forward p̂A! := p̂!(A(p) ∪ . . . ) : Ĥ(W,Q) → Ĥ(B, Q). One of our main results lifts the cohomological version of the Atiyah-Singer index theorem to smooth cohomology. It states that p̂A! ◦ ĉh = ĉh ◦ p̂!.
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