transversally elliptic operator, fundamental class, K-homology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:57:00Z http://mathoverflow.net/feeds/question/69595 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69595/transversally-elliptic-operator-fundamental-class-k-homology transversally elliptic operator, fundamental class, K-homology Yanli Song 2011-07-06T01:25:03Z 2011-07-06T02:10:43Z <p>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.</p> <p>And what if M is no longer compact?</p> http://mathoverflow.net/questions/69595/transversally-elliptic-operator-fundamental-class-k-homology/69596#69596 Answer by Alain Valette for transversally elliptic operator, fundamental class, K-homology Alain Valette 2011-07-06T02:10:43Z 2011-07-06T02:10:43Z <p>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"?</p> <p>Note: your title mentions the fundamental class, but your question doesn't.</p>