** Betti numbers **

Started with M. Atiyah (1976) for manifolds. Very soon (1979), Connes obtained a version for foliations. In 1986, Cheeger and Gromov extended the notion to arbitrary countable groups.

**1. Von Neumann dimension **

Murray-von Neumann 1944. Let be a closed -invariant subspace in . A real number can be defined in such of way that

- .
- .
- If is bounded and equivariant, .

** 1.1. -trace **

Let be an operator which commutes with the left regular representation. Express it in the canonical basis . The diagonal elements are all equal. *Denote* this value by . Check that .

** 1.2. Case **

Orthogonal projection onto is an element which commutes with the left regular representation. So define .

Since is an orthogonal projector,

Therefore implies , and thus , .

** 1.3. Exercise **

If , Fourier transform maps to , the action is by multiplication with function . -equivariant operators are multiplication operators with bounded functions . corresponds to constant function 1. . Orthogonal projections are indicator functions of Borel sets , invariant subspaces are of the form , measure of .

** 1.4. General case **

View projector on as a matrix in block form. Diagonal blocks turn out to be -equivariant. Define .

** 1.5. Homology **

Let be a countable simplicial complex with free cocompact -action. chains make sense, therefore -homology is defined. Reduced -homology is isomorphic to the subspace of harmonic -chains . This can be viewed as a -invariant subspace in finitely many copies of . Define

Observe that

Proposition 1For all -connected , the -Betti numbers are the same up to . This defines .

** 1.6. Cheeger-Gromov’s definition **

For a general group , an infinite dimensional contractible simplicial complex (with free action of ) may be needed. Exhaust it with co-finite ones, and take a limit of . Does not depend on exhaustion.

**2. Some values of Betti numbers **

Finite groups: all 0 but the first, .

Infinite groups: .

Free group : .

generated by elements: .

infinite amenable: (I will prove this).

Surface group of genus : .

Lattices in , ,…: exactly one nonzero Betti number.

** 2.1. Some results **

Passing to finite index subgroup multiplies Betti numbers by index.

Euler characteristic .

Lück’s approximation theorem: see lecture nr 3.

** 2.2. Atiyah’s conjecture **

Atiyah: if torsion free, are Betti numbers integers ?

The generalized form (under restrictions on torsion) has been disproved, at least for actions. Grigorchuk-Zuk calculation for the lamplighter group.