## Notes of Dima Shlyakhtenko’s Cambridge lecture 25-01-2017

Homology ${L^2}$-Betti numbers for subfactors and quasiregular inclusions

Joint with Sorin Popa and Stefan Vaes.

There were several competing definitions, and they turn out to be equivalent, relief.

1. ${L^2}$ Betti numbers

Associative algebras ${A}$ (and bimodules ${V}$) have a Hochschild cohomology defined as follows. Cochains are maps ${A^{\otimes k}\rightarrow V}$. Coboundary is such that 1-cocycles are derivations. If ${A}$ is augmented (e.g. a group algebra), right ${A}$-modules can be turned into bimodules by using the augmentation ${\epsilon}$ as a left action.

When ${A={\mathbb C}\Gamma}$ is a group algebra, ${\Gamma=\pi_1(X)}$ and ${\tilde X}$ is contractible, then ${H^\cdot({\mathbb C}\Gamma,{\mathbb C})=H^\cdot(X)}$. Another choice of coefficients is ${\ell^2(\Gamma)}$, (or its completion ${L(\Gamma)}$). Then ${H^\cdot(\Gamma,\ell^2(\Gamma))}$ are modules over ${L(\Gamma)}$, hence have a dimension, the Betti number

$\displaystyle \begin{array}{rcl} \beta_k^{(2)}(\Gamma)=\mathrm{dim}_{L(\Gamma)}H^k(\Gamma,\ell^2(\Gamma)). \end{array}$

${L^2}$-Betti numbers of infinite groups are additive under free products and multiplicative under direct products.

A factor has some grouplike symmetry. To embody it, we have two choices, ${C^*}$-tensor categories and ${M\otimes M^{op}}$ included in Popa’s symmetric enveloping algebra.

2. The ${C^*}$-tensor categories point of view

Let ${\mathcal{C}}$ be a tensor category. When ${\alpha}$, ${\beta}$ are irreducible objects, we denote by ${(\alpha,\beta)}$ the set of intertwiners.

The tube algebra is defined, as a vectorspacen by

$\displaystyle Tube(\mathcal{C})=\bigoplus_{i,i,\alpha\in Irr(\mathcal{C})}(i\alpha,\alpha j).$

The multiplication ${\mathcal{A}_{ij}\times\mathcal{A}_{jk}\rightarrow\mathcal{A}_{ik}}$ is

$\displaystyle \begin{array}{rcl} VW=(V\otimes 1)(1\otimes W). \end{array}$

Define the idempotent ${p_i=id_{ie,ei)}\in\mathcal{A}_{ii}}$, where ${e}$ is the trivial representation. They generate a subalgebra ${\mathcal{B}}$. The augmentation ${\epsilon:\mathcal{A}\rightarrow{\mathbb C}}$ is defined by

$\displaystyle \begin{array}{rcl} \epsilon(p_i)=0,~(i\not=e),\quad \epsilon(id_\alpha)=d(\alpha). \end{array}$

We think of as a representation of ${\mathcal{A}}$ as a “representation” of ${\mathcal{C}}$.

The cohomology of ${\mathcal{A}}$ and a right ${\mathcal{A}}$-module ${K}$ is defined as follows. Cochains are morphisms ${p_e\mathcal{A}^{\otimes_\mathcal{B}(n+1)}}$ to ${K}$. The coboundary is given by the same formula as Hochschild.

Fot ${n=1}$, ${H^1}$ corresponds to derivations ${D}$ ${D(VW)=D(V)W+\epsilon(V)D(W)}$, modulo inner derivations ${D_\xi(V)=\epsilon(V)\xi-\xi V}$.

Possible choices for ${K}$: ${\epsilon}$ yields cohomology ${L^2(\mathcal{A},Trace)}$ leads to ${L^2}$ Betti numbers.

3. The quasiregular inclusion point of view

Let ${T\subset S}$ be a quasiregular inclusion. Define the tube algebra by a similar formula for the category ${_T S_T}$. It is Morita equivalent to the preceding tube algebra.

Cochains involve tensor powers of ${S}$ over ${T}$.

In case of trivial coefficients, i.e. take module ${S}$. Then get ${H_k(T\subset S,S)=0}$ if ${T}$ has finite index in ${S}$. This had been observed by Jones. More genrally, ${H_\cdot(\mathcal{A}_{\mathcal{C}},\epsilon)=0}$ if ${\mathcal{C}}$ has finite depth.

For calculations, one replaces cocycles ${Z}$ and coboundaries ${B}$ with smaller subspaces.

For instance, for amenable categories (in a Folner sense), Betti numbers vanishes. For finite index inclusions, ${\beta_0^{(2)}=1/[S:T]}$. Additivity under free products and multiplicativity under direct products od categories holds.

3.1. Graphical picture for trivial coefficients

On a 2-sphere with punctures, draw cochains as bodies with as many legs as factors in a tensor product. Then coboundary is alternating sum of picture obtained by removing each puncture at a time.

4. Computations

Mystery: what is ${H_k(T\subset T;}$ trivial coefficients ${)}$ ?

Kyed, Raum, Vaes, Valvekans have a recent result.

One may conjecture that for the category of quantum groups, one recovers Betti numbers of quantum groups. The study of fusion rings seems to indicate that it could be wrong.

One recovers Gaboriau’s Betti numbers for group ${\Gamma}$ acting on space ${X}$, ${T=L^{\infty}(X)}$ but ${S}$ is not quite ${L^{\infty}(X)\times\Gamma}$.