Hilbert-Smith conjecture
Encyclopedia
In mathematics
, the Hilbert–Smith conjecture is concerned with the transformation groups of manifold
s; and in particular with the limitations on topological group
s G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action
on M, it states that G must be a Lie group
.
Because of known structural results on G, it is enough to deal with the case where G is the additive group Zp of p-adic integers, for some prime number
p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold.
The naming of the conjecture is for David Hilbert
, and the American topologist Paul A. Smith
. It is considered by some to be a better formulation of Hilbert's fifth problem
, than the characterisation in the category of topological group
s of the Lie group
s often cited as a solution.
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 Hilbert–Smith conjecture is concerned with the transformation groups 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; and in particular with the limitations on topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group 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...
on M, it states that G must be a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
.
Because of known structural results on G, it is enough to deal with the case where G is the additive group Zp of p-adic integers, for some prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p. An equivalent form of the conjecture is that Zp has no faithful group action on a topological manifold.
The naming of the conjecture is for David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, and the American topologist Paul A. Smith
Paul A. Smith
Paul Althaus Smith was an American mathematician. His name occurs in two significant conjectures in geometric topology: the Smith conjecture, which is now a theorem, and the Hilbert-Smith conjecture, still open...
. It is considered by some to be a better formulation of Hilbert's fifth problem
Hilbert's fifth problem
Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...
, than the characterisation in the category of topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s of the Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s often cited as a solution.