MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of differential forms on $X$. On the other hand, it is intuitively clear, that $\Omega^\ast(E)$ contains all the information on $E$ one needs to describe all the rational invariants of $E$, including its characteristic classes. So, I wonder, if there is a homotopy-invariant way to extract this information? That is, I need a method, non-involving the usual Chern-Weil formalism, non-appealing to the general linear group etc., a method, roughly speaking, applicable to any differential graded algebra or even a dg module over $\Omega^\ast(X)$. In addition, it is interesting, how much of the actual structure of $\Omega^\ast(E)$ one actually needs: is it enough to consider the structure of differential graded module over $\Omega^\ast(X)$ on it?

This brings us to a more general question: is there a good definition of (algebraic) K-theory of differential graded modules/algebras over a differential graded algebra, which would reproduce the usual K-theory of a smooth manifold, if applied to its algebra of differential forms? Of course, we assume, that the class of the module $\Omega^\ast(E)$ should correspond to the class of $E$ in $K^0(X)$. The theory we ask for should be invariant under the homotopy equivalencies of algebras. I believe, the answer to this question should exist somewhere, so, if you know a reference, please, do send it to me.

share|cite|improve this question
Have you looked at Karoubi's "Homologie cyclique et K-th\'eorie" (Asterisque 149, Soc. Math. France, 1987)? There might be something in this direction... – Alain Valette Dec 8 '11 at 14:43
The 'standard' way of defining the $K$-theory of a DG-algebra $A$ is as the Waldhausen $K$-theory of the category perfect $A$-modules, but you don't recover the $K$-theory of $M$ when $A=\Omega^*(M)$. Notice that $\Omega^*(M)$ only encodes the real homotopy type of $M$, while the $K$-theory of $M$ keeps more subtle information. – Fernando Muro Dec 8 '11 at 17:41
Well, of course, it would be preferrable to obtain all of $K(M)$ from this kind of theory, but real homotopy type will be enough for the time-being. Can you give a reference to a paper dealing with this subject? I mean the Waldhausen $K$-theory of the category of perfect modules? – gshar Dec 8 '11 at 19:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.