Euler characteristic and the class of unit in K-theory
Let \(\chi (\xi_{\phi})\) be the Euler characteristic of the \(S^ 1\)- bundle over a compact Riemannian surface \(M_ g\) of genus \(g\geq 2\) associated to an arbitrary homomorphism \(\phi\) from \(\pi_ 1(M_ g)\) into the group \(Top_+(S^ 1)\) of all orientation preserving homeomorphisms of \(S^ 1\). Then, in the \(K_ 0\)-group of the reduced crossed product \(C(S^ 1)\rtimes \pi_ 1(M_ g)\), the class of unit [1] is a torsion element with \(\chi (\xi_{\phi})[1]=0\).
Euler characteristic of an \(S^ 1\)-bundle over a compact Riemannian surface
\(K_ 0\)-group of reduced crossed product
scientific article; zbMATH DE number 3956004
scientific article