Atiyah conjecture

In 1976, Michael Atiyah introduced ${\displaystyle l^{2}}$-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 ${\displaystyle l^{2}}$-Betti numbers as von Neumann dimensions of the resulting ${\displaystyle l^{2}}$-cohomology groups, and computed several examples, which all turned out to be rational numbers. He therefore asked if it is possible for ${\displaystyle l^{2}}$-Betti numbers to be irrational.

Since then, various researchers asked more refined questions about possible values of ${\displaystyle l^{2}}$-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 ${\displaystyle l^{2}}$-Betti numbers are integers.

The most general question open as of late 2011 is whether ${\displaystyle l^{2}}$-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 ${\displaystyle l^{2}}$-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 (2013). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society. 107 (6): 1424–1448. arXiv:0909.2360. doi:10.1112/plms/pdt029.

• Eckmann, Beno (2000). "Introduction to ${\displaystyle \ell _{2}}$-methods in topology: reduced ${\displaystyle \ell _{2}}$-homology, harmonic chains, ${\displaystyle \ell _{2}}$-Betti numbers". Israel Journal of Mathematics. Vol. 117. pp. 183–219. doi:10.1007/BF02773570.