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

How transversally elliptic pseudo-differential operator naturally induces a K-homology class in KK(A, C), where the algebra A is the crossed product algebra C(M) ⋊ G, where M is compact manifold and G is compact Lie group. Do you have any reference paper about this work? Thanks.

And what if M is no longer compact?

share|cite|improve this question
up vote 4 down vote accepted

Since $G$ is compact, averaging over $G$ you may assume that your operator is $G$-invariant. If you assume that $G$ acts freely on your manifold, then $C_0(M)\rtimes G$ is Morita equivalent to $C_0(M/G)$, and what you want boils down to the standard fact that an elliptic pseudo-differential operator defines a $K$-homology class on a compact manifold. You also see that it does not work any longer when $M$ is not compact. For the general case, have you checked in Atiyah's Springer Lecture Notes 401, "Elliptic operators and compact groups"?

Note: your title mentions the fundamental class, but your question doesn't.

share|cite|improve this answer

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.