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.

We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors $\Gamma(\mu,m)"$ up to isomorphism. We do this by showing that free Araki-Woods factors $\Gamma(\mu, m)"$ arising from finite symmetric Borel measures $\mu$ on $\mathbf{R}$ whose atomic part $\mu_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})$ as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.