Hilbert's fifth problem
Encyclopedia
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 group
s. The theory of Lie groups describes continuous symmetry
in mathematics; its importance there and in theoretical physics
(for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory
and the theory of topological manifold
s. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed?
The expected answer was in the negative (the classical group
s, the most central examples in Lie group theory, are smooth manifolds). This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.
s that were also topological manifold
s. In terms closer to those that Hilbert would have used, near the identity element
e of the group G in question, we have some open set
U in Euclidean space
containing e, and on some open subset V of U we have a continuous mapping
that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group
. The problem is then to show that F is a smooth function
near e (since topological groups are homogeneous space
s, they look the same everywhere as they do near e).
Another way to put this is that the possible differentiability class of F doesn't matter: the group axioms collapse the whole Ck gamut.
in 1933 , for compact group
s. The locally compact abelian group case was solved in 1934 by Lev Pontryagin. The final resolution, at least in this interpretation of what Hilbert meant, came with the work of Andrew Gleason
, Deane Montgomery
and Leo Zippin
in the 1950s.
In 1953, Hidehiko Yamabe
obtained the final answer to Hilbert’s Fifth Problem: a connected locally compact group is a projective limit of a sequence of Lie groups, and if "has no small subgroups" (a condition defined below), then G is a Lie group. However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.
More generally, every locally compact, almost connected group is the projective limit of a Lie group. If we consider a general locally compact group and the connected component of the identity , we have a group extension
As a totally disconnected group / has an open compact subgroup, and the pullback of such an open compact subgroup is an open, almost connected subgroup of . In this way, we have a smooth structure on , since it is homeomorphic to , where is a discrete set.
s. A topological group G, or a partial piece of a group like F above, is said to have no small subgroups if there is a neighbourhood N of e containing no subgroup bigger than {e}. For example the circle group satisfies the condition, while the p-adic integers Zp as additive group
does not, because N will contain the subgroups
for all large integers k. This gives an idea of what the difficulty is like in the problem. In the Hilbert–Smith conjecture case it is a matter of a known reduction to whether Zp can act faithfully on a closed manifold
. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact group
s, as those having no small subgroups.
discuss the thesis of Per Enflo, on Hilbert's fifth problem without compactness.
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 concerns the characterization of 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. The theory of Lie groups describes continuous symmetry
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
in mathematics; its importance there and in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
(for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
and the theory of topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
s. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed?
The expected answer was in the negative (the classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s, the most central examples in Lie group theory, are smooth manifolds). This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.
Classic formulation
A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groupTopological 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 that were also topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
s. In terms closer to those that Hilbert would have used, near the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
e of the group G in question, we have some open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
U in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
containing e, and on some open subset V of U we have a continuous mapping
that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group
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....
. The problem is then to show that F is a smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
near e (since topological groups are homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s, they look the same everywhere as they do near e).
Another way to put this is that the possible differentiability class of F doesn't matter: the group axioms collapse the whole Ck gamut.
Solution
The first major result was that of John von NeumannJohn von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
in 1933 , for compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
s. The locally compact abelian group case was solved in 1934 by Lev Pontryagin. The final resolution, at least in this interpretation of what Hilbert meant, came with the work of Andrew Gleason
Andrew Gleason
Andrew Mattei Gleason was an American mathematician and the eponym of Gleason's theorem and the Greenwood–Gleason graph. After briefly attending Berkeley High School he graduated from Roosevelt High School in Yonkers, then Yale University in 1942, where he became a Putnam Fellow...
, Deane Montgomery
Deane Montgomery
Deane Montgomery was a mathematician specializing in topology who was one of the contributors to the final resolution of Hilbert's fifth problem in the 1950s. He served as President of the American Mathematical Society from 1961 to 1962....
and Leo Zippin
Leo Zippin
Leo Zippin was an American mathematician. His parents were Bella Salwen and Max Zippin who immigrated to New York City from the Ukraine in 1903.-Education:...
in the 1950s.
In 1953, Hidehiko Yamabe
Hidehiko Yamabe
-External links:...
obtained the final answer to Hilbert’s Fifth Problem: a connected locally compact group is a projective limit of a sequence of Lie groups, and if "has no small subgroups" (a condition defined below), then G is a Lie group. However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.
More generally, every locally compact, almost connected group is the projective limit of a Lie group. If we consider a general locally compact group and the connected component of the identity , we have a group extension
As a totally disconnected group / has an open compact subgroup, and the pullback of such an open compact subgroup is an open, almost connected subgroup of . In this way, we have a smooth structure on , since it is homeomorphic to , where is a discrete set.
Alternate formulation
Another view is that G ought to be treated as a transformation group, rather than abstractly. This leads to the formulation of the Hilbert–Smith conjecture, unresolved .No small subgroups
An important condition in the theory is no small subgroupNo small subgroup
In mathematics, especially in topology, a topological group G is said to have no small subgroup if there exists a neighborhood U of the identity that contains no nontrivial subgroup of G. An abbreviation '"NSS"' is sometimes used...
s. A topological group G, or a partial piece of a group like F above, is said to have no small subgroups if there is a neighbourhood N of e containing no subgroup bigger than {e}. For example the circle group satisfies the condition, while the p-adic integers Zp as additive group
Additive group
An additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
does not, because N will contain the subgroups
for all large integers k. This gives an idea of what the difficulty is like in the problem. In the Hilbert–Smith conjecture case it is a matter of a known reduction to whether Zp can act faithfully on a closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact group
Locally compact group
In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
s, as those having no small subgroups.
Infinite dimensions
Researchers have also considered Hilbert's fifth problem without supposing finite dimensionality. The last chapter of Benyamini and LindenstraussJoram Lindenstrauss
Joram Lindenstrauss is an Israeli mathematician working in functional analysis. He is professor emeritus of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel.-Biography:...
discuss the thesis of Per Enflo, on Hilbert's fifth problem without compactness.