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, .
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 .
Orthogonal projection onto is an element which commutes with the left regular representation. So define .
Since is an orthogonal projector,
Therefore implies , and thus , .
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 .
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
Proposition 1 For 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.