In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of

l^2

History

In 1976,

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...

introduced

l^2

-cohomology of manifolds with a free co-compact

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of a discrete

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

(e.g. the

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

of a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

manifold together with the action of the fundamental group by deck transformations.) Atiyah defined also numbers as von Neumann dimensions of the resulting groups, and computed several examples, which all turned out to be

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

s. He therefore asked if it is possible for

l^2

-Betti numbers to be

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

. Since then, various researchers asked more refined questions about possible values of

l^2

-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 In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...

, then the

l^2

-Betti numbers are

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s. The most general question open as of late 2011 is whether

l^2

-Betti numbers are rational if there is a bound on the orders of

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s 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 group In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A ...

s, 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 Timothy Austin (born April 14, 1971) is an American former professional boxer. He is now a coach at the Cincinnati Golden Gloves gym in Cincinnati. Amateur career Austin had an outstanding amateur career, compiling a record of 113-9. Amateu ...

showed in 2009 that

l^2

-Betti numbers can assume transcendental values. Later it was shown that in that case they can be any non-negative

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

References

* * *{{Cite news , last = Eckmann , first = Beno , title = Introduction to

\ell_2

-methods in topology: reduced

\ell_2

-homology, harmonic chains,

\ell_2

-Betti numbers , journal = Israel Journal of Mathematics , volume = 117 , year = 2000 , pages = 183–219 , doi=10.1007/BF02773570 , doi-access=free Conjectures Cohomology theories Differential geometry Differential topology