In 1976 Michael Atiyah introduced -cohomology of manifolds with a free co-compact action of a discrete countable group (e.g. the universal cover of a compact manifold together with the action of the fundamental group by deck transformations.) Atiyah defined also -Betti numbers as von Neumann dimensions of the resulting -cohomology groups, and computed several examples, which all turned out to be rational numbers. He therefore asked if it is possible for -Betti numbers to be irrational.
Since then, various researchers asked more refined questions about possible values of -Betti numbers, all of which are customarily referred to as "Atiyah conjecture".
Many positive results were proven by Peter Linnell. For example, if the group acting is a free group, then the -Betti numbers are integers.
The most general question open as of late 2011 is whether -Betti numbers are rational if there is a bound on the orders of finite subgroups of the group which acts. In fact, a precise relationship between possible denominators and the orders in question is conjectured; in the case of torsion-free groups this statement generalizes the zero-divisors conjecture. For a discussion see the article of B. Eckmann.
In the case there is no such bound, Tim Austin showed in 2009 that -Betti numbers can assume transcendental values. Later it was shown that in that case they can be any non-negative real numbers.
- Atiyah, M. F (1976). "Elliptic operators, discrete groups and von Neumann algebras". Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974). Paris: Soc. Math. France. pp. 43–72. Astérisque, No. 32–33.
- Austin, Tim (2009-09-12). "Rational group ring elements with kernels having irrational dimension". arXiv:0909.2360.
- Eckmann, Beno (2000). "Introduction to l_2-methods in topology: reduced l_2-homology, harmonic chains, l_2-Betti numbers". Israel J. Math. 117. pp. 183–219.