In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of
-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.)
-Betti numbers as von Neumann dimensions of the resulting
-cohomology groups, and computed several examples, which all turned out to be rational numbers.
Since then, various researchers asked more refined questions about possible values of
-Betti numbers, all of which are customarily referred to as "Atiyah conjecture".
-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.
In the case there is no such bound, Tim Austin showed in 2009 that