Bastien Jean

The aim of this talk is to explain the $L^2$-index theorem of Atiyah. Consider a Riemannian manifold $\tilde{M}$ endowed with a free and proper action of a discrete group $\Gamma$ with compact quotient $M:=\tilde{M} /\Gamma$, we are interested in the study of an elliptic differential operator $\tilde{D}$ between hermitian vector bundle on $\tilde M$ obtained as the lifting of a differential operator $D$ on $M$. If $\Gamma$ is finite, the usual index theorem gives us $index(\tilde{D}) = |\Gamma|\cdot index(D)$. The Atiyah index theorem is a generalisation including the case of infinite covering, however in this setting the index of $\tilde D$ is not well defined due to the non-compactness of $\tilde M$ and to solve this problem we will first need to introduce the notion of $\Gamma$-dimension defined by Von Neumann.