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We introduce   $L^2$  -Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II  $_1$   factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups   $\Gamma$  , we recover the ordinary (co)homology of   $\Gamma$  . For Cartan subalgebras, we recover Gaboriau’s   $L^2$  -Betti numbers for the associated equivalence relation. In this common framework, we prove that the   $L^2$  -Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property, and amenability. We compute the   $L^2$  -Betti numbers for the standard invariants of the Temperley–Lieb–Jones subfactors and of the Fuss–Catalan subfactors, as well as for free products and tensor products.

Cohomology and $L^2$ -Betti Numbers for Subfactors and Quasi-Regular Inclusions