$L^2$-Betti Numbers of Infinite Configuration Spaces

Abstract

The space ${\Gamma }_X$ of all locally finite configurations in a infinite covering $X$ of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over ${\Gamma }_X$, equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.