Bunke, Ulrich and Schick, Thomas (2009) SMOOTH K-THEORY. ASTERISQUE (328). pp. 45-135. ISSN 0303-1179,
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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 (p) over cap! : (K) over cap (W) -> (K) over cap (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 c (h) over cap : (K) over cap (B) -> (H) over cap (B,Q), where (H) over cap (B, Q) denotes the smooth extension of rational cohomology, and we show that c (h) over cap induces a rational isomorphism. If p: W -> B is a proper submersion with a smooth K-orientation, then we define a class A(p) is an element of (H) over cap (ev) (W,Q) (see Lemma 6.17) and the modified push-forward (p) over cap (A)(!) := (p) over cap!(A(p) boolean OR ...) : (H) over cap (W,Q) -> (H) over cap (B, Q). One of our main results lifts the cohomological version of the Atiyah-Singer index theorem to smooth cohomology. It states that (p) over cap (A)(!) circle c (h) over cap = c (h) over cap circle (p) over cap!(.)
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | ; Deligne cohomology; smooth K-theory; Chern character; families of elliptic operators; Atiyah-Singer index theorem |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Ulrich Bunke |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 12 Oct 2020 05:15 |
| Last Modified: | 12 Oct 2020 05:15 |
| URI: | https://pred.uni-regensburg.de/id/eprint/29644 |
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