In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
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
-cohomology of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s 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
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
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 ration ...
s. He therefore asked if it is possible for
-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
-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
-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 language ...
s.
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
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
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
d; 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 boxing, 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.
...
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 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 real ...
s.
References
*
*
*{{Cite news
, last = Eckmann
, first = Beno
, title = Introduction to
-methods in topology: reduced
-homology, harmonic chains,
-Betti numbers
, journal =
Israel Journal of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press).
Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the jou ...
, volume = 117
, year = 2000
, pages = 183–219
, doi=10.1007/BF02773570 , doi-access=free
Conjectures
Cohomology theories
Differential geometry
Differential topology