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We introduce $L^2$-Betti numbers, as well as a general homology andcohomology theory for the standard invariants of subfactors, through theassociated quasi-regular symmetric enveloping inclusion of II_1 factors. Weactually develop a (co)homology theory for arbitrary quasi-regular inclusionsof von Neumann algebras. For crossed products by countable groups $\Gamma$, werecover the ordinary (co)homology of $\Gamma$. For Cartan subalgebras, werecover Gaboriau's $L^2$-Betti numbers for the associated equivalence relation.In this common framework, we prove that the $L^2$-Betti numbers vanish foramenable inclusions and we give cohomological characterizations of property(T), the Haagerup property and amenability. We compute the $L^2$-Betti numbersfor the standard invariants of the Temperley-Lieb-Jones subfactors and of theFuss-Catalan subfactors, as well as for free products and tensor products.

Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions