Holmstrom Multiplicative K-theory

Multiplicative K-theory

Karoubi: “In this paper we introduce a new type of K-theory, called ”multiplicative K-theory“ kn(A) for A a Frechet algebra, which is intermediary between algebraic K-theory, denoted Kn(A), and topological K-theory, denoted Kntop(A). This new theory is computable in terms of Kntop(A) and cyclic homology HC(A). The homomorphism from Kn(A) to kn(A) we define in the paper detects all known primary and secondary characterictic classes from algebraic K-theory to cyclic homology (for example the Borel regulator if A = the field of complex numbers). It is related to the multiplicative character of a Fredholm module defined previously by A. Connes and the author.“

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Multiplicative K-theory

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Multiplicative K-theory

KT (K-theory), NCG (Algebra and noncommutative geometry)?

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Multiplicative K-theory

MR1076525 (92a:55006) Karoubi, Max(F-PARIS7) Théorie générale des classes caractéristiques secondaires. (French. English summary) [General theory of secondary characteristic classes] $K$-Theory 4 (1990), no. 1, 55–87. 55N15 (14F40 19D55 19L10 46L80) PDF Doc Del Clipboard Journal Article Make Link

In this paper, the author describes a general method using $K$-theory to define primary characteristic classes (like Chern classes) or secondary characteristic classes (like the Godbillon-Vey invariant or regulators). To this end, he defines a new functor $MK(X)$, called multiplicative $K$-theory of $X$, which is the target of known primary and secondary characteristic classes of vector bundles with additional structures. More precisely, $MK(X)$ is the group associated to the monoid of isomorphism classes of multiplicative vector bundles. The tensor product induces a ring structure on $MK(X)$. The author establishes an important exact sequence which relates multiplicative $K$-theory, topological $K$-theory and \n de Rham\en cohomology. He also describes concisely applications of this theory to the case of algebraic and foliated varieties. The Chern-Weil homomorphism is discussed in detail, in both the analytical and simplicial case. Unstable and stable multiplicative $K$-theories are then defined. At the end, it is shown how multiplicative $K$-theory is related to cyclic homology.

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For a survey level to cohomology of foliations esp with emphasis on generalizations of Godbillon-Vey what would you recommend? thanks, jim

In answer to your question, let me mention two papers of mine where secondary classes are expressed with great details in more general situations (not only foliations, but also holomorphic or algebraic bundles, regulators, etc.) They are more conveniently expressed in terms of (multiplicative) K-theory, but it is easy to translate them into classes in (multiplicative) cohomology via the Chern character.

This multiplicative cohomology is defined for a suitable filtration of the de Rham complex and is a refined version of Deligne cohomology.

See the URL:

http://people.math.jussieu.fr/~karoubi/Publications.html

(Publications Nr 43 and 45)

Best wishes, Max

nLab page on Multiplicative K-theory