Lecture Series

Speaker: Thomas Schick

Title: Lecture 4 on L^2-Betti numbers, their applications and conjectures about them

Abstract:

Betti numbers are among the most fundamental, and extremely useful,
  invariants of topological spaces.

  However, for non-compact spaces, in particular those with a lot of symmetry,
  they are typically not useful, because they are typically infinite.

  $L^2$-Betti numbers are a very successfull replacement, in particular for
  manifolds or cell complexes $X$ with a free action of a discrete group
  $\Gamma$. There definition is based on the use of Hilbert spaces with a
  (good) $\Gamma$-action and a dimension function for those, the von Neumann
  dimension. One then uses chain complexes and cohomology with square summable
  (co)cycles, which for exactly the kind Hilbert spaces for which the dimension
  function is defined.

  That way, the von Neumann dimension of this $L^2$-cohomology gives by
  definition the $L^2$-Betti numbers.

  The course will introduce (just as much as is needed) in the underlying
  (analytic) theory and introduce the $L^2$-Betti numbers (and related
  invariants). The main part is then devoted to the study of their properties,
  the computation of examples and the application of these calculations to
  question outside the theory of $L^2$-invariants. These applications range
  from differential geometry via topology to the theory of groups and group
  rings. 

There are many open questions and conjectures around these invariants, with
constant progress over the years (special cases,\ldots). We will introduce
some of them, e.g.~the Atiyah question about the possible values of
$L^2$-Betti numbers, and report on their current status.

Recently, there are attempts (in special cases successfull) to define variants
of $L^2$-Betti numbers with finite field coefficients (non-obvious, as there
is no good concept of Hilbert space available in this context); we will get to
this toward the end of the lecture series.