“Atiyah conjecture”的意思、由来-开放百科全书

词条

Atiyah conjecture

释义

History
Results
References

{{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 -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, 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.

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
}}

4 : Conjectures|Cohomology theories|Differential geometry|Differential topology

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。