Geometric bivariant K-theory and K-theoretical virtual fundamental classes

Motivated by geometric cycles (D-branes in string theor), I will present a framework for geometric bivariant K-theory KK(X, Y) for any pair of smooth manifolds (orbifolds). When $X$ is a point, $KK(X, Y)$ is isomorphic to K-theory of $Y$. When $Y$ is a point, $KK(X, Y)$ is isomorphic to K-homology of $X$. As an application, K-theoretical fundamental class will be interpreted as a bivariant cycle between $M_{g, k}$ and $X^k$. Localisation of K-theoretical virtual fundamental classes will be discussed in this framework. This talk is based on joint work with Bohui Chen and Jianxun Hu.