In Mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.

History

In 1976 Michael Atiyah

Michael Atiyah

Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

 introduced -cohomology of manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s with a free co-compact action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of a discrete countable group (e.g. the universal cover of a compact manifold together with the action of the fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 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".

Results

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, 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 transcendal values. Later it was shown that in that case they can be any non-negative real numbers.