Atiyah conjecture

In Mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of ${\displaystyle l^{2}}$-Betti numbers.

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.

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