In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.
In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.
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.
Sir Michael Francis Atiyah was a British-Lebanese mathematician specialising in geometry. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004.
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