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{{for |the conjecture about sets of points in Euclidean space|Atiyah conjecture on configurations}} | |||
In [[Mathematics]], the '''Atiyah conjecture''' is a collective term for a number of statements about restrictions on possible values of [[L2 cohomology|<math>l^2</math>-Betti numbers]]. | |||
==History== | |||
In 1976 [[Michael Atiyah]] introduced [[L2 cohomology|<math> l^2</math> -cohomology]] of [[manifold]]s with a free co-compact [[group action|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 <math>l^2</math>-Betti numbers as [[von Neumann dimension]]s of the resulting <math>l^2</math>-cohomology groups, and computed several examples, which all turned out to be rational numbers. He therefore asked if it is possible for <math>l^2</math>-Betti numbers to be [[irrational numbers|irrational]]. | |||
Since then, various researchers asked more refined questions about possible values of <math>l^2</math>-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 <math>l^2</math>-Betti numbers are integers. | |||
The most general question open as of late 2011 is whether <math>l^2</math>-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 [[group ring#Group rings over an infinite group|zero-divisors conjecture]]. For a discussion see | |||
the article of B. Eckmann. | |||
In the case there is no such bound, [[Tim Austin (mathematician)|Tim Austin]] showed in 2009 that <math>l^2</math>-Betti numbers can assume transcendal values. Later it was shown that in that case they can be any non-negative real numbers. | |||
==References== | |||
* {{Cite book| publisher = Soc. Math. France | |||
| pages = 43–72. Astérisque, No. 32–33 | |||
| last = Atiyah | |||
| first = M. F | |||
| title = Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974) | |||
| chapter = Elliptic operators, discrete groups and von Neumann algebras | |||
| location = Paris | |||
| year = 1976 | |||
}} | |||
* {{Cite arXiv | |||
| last = Austin | |||
| first = Tim | |||
| title = Rational group ring elements with kernels having irrational dimension | |||
| date = 2009-09-12 | |||
| eprint = 0909.2360 | |||
}} | |||
*{{ Cite news | |||
| last = Eckmann | |||
| first = Beno | |||
| title = Introduction to l_2-methods in topology: reduced l_2-homology, harmonic chains, l_2-Betti numbers | |||
| journal = Israel J. Math. | |||
| volume = 117 | |||
| year = 2000 | |||
| pages = 183–219 | |||
}} | |||
[[Category:Conjectures]] | |||
[[Category:Cohomology theories]] | |||
[[Category:Differential geometry]] | |||
[[Category:Differential topology]] | |||
{{topology-stub}} | |||
Latest revision as of 11:32, 28 November 2013
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In Mathematics, 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 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".
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.
References
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