Kuiper's theorem
Encyclopedia
In mathematics
, Kuiper's theorem (after Nicolaas Kuiper
) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space
H. It states that the space
GL(H) of invertible bounded
endomorphisms H is such that all maps from any finite complex
Y to GL(H) are homotopic to a constant, for the norm topology on operators.
A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible
, ie. all its homotopy group
s are trivial. This result has important uses in topological K-theory
.
and not at all contractible. In fact it is homotopy equivalent to its maximal compact subgroup
, the unitary group
U of H. The proof that the complex general linear group and unitary group have the same homotopy type is by the Gram-Schmidt process, or through the matrix polar decomposition, and carries over to the infinite-dimensional case of separable Hilbert space, basically because the space of upper triangular matrices is contractible as can be seen quite explicitly. The underlying phenomenon is that passing to infinitely many dimensions causes much of the topological complexity of the unitary groups to vanish; but see the section on Bott's unitary group, where the passage to infinity is more constrained, and the resulting group has non-trivial homotopy groups.
, sometimes denoted S∞, in infinite-dimensional Hilbert space
H is a contractible space
, while no finite-dimensional spheres are contractible. This result, certainly known decades before Kuiper's, may have the status of mathematical folklore, but it is quite often cited. In fact more is true: S∞ is diffeomorphic to H, which is certainly contractible by its convexity. One consequence is that there are smooth counterexamples to an extension of the Brouwer fixed-point theorem to the unit ball in H. The existence of such counter-examples that are homeomorphism
s was shown in 1943 by Shizuo Kakutani
, who may have first written down a proof of the contractibility of the unit sphere. But the result was anyway essentially known (in 1935 Andrey Nikolayevich Tychonoff
showed that the unit sphere was a retract of the unit ball).
The result on the group of bounded operators was proved by the Dutch mathematician Nicolaas Kuiper
, for the case of a separable Hilbert space; the restriction of separability was later lifted. The same result, but for the strong operator topology
rather than the norm topology, was published in 1963 by Jacques Dixmier
and Adrien Douady
. The geometric relationship of the sphere and group of operators is that the unit sphere is a homogeneous space
for the unitary group U. The stabiliser of a single vector v of the unit sphere is the unitary group of the orthogonal complement of v; therefore the homotopy long exact sequence predicts that all the homotopy groups of the unit sphere will be trivial. This shows the close topological relationship, but is not in itself quite enough, since the inclusion of a point will be a weak homotopy equivalence only, and that implies contractibility directly only for a CW complex
. In a paper published two years after Kuiper's, Richard Palais provided technical results on infinite-dimensional manifolds sufficient to resolve this issue.
applies. It is certainly not contractible. The difference from Kuiper's group can be explained: Bott's group is the subgroup in which a given operator acts non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis {ei}, for some N, being the identity on the remaining basis vectors.
The result on the contractibility of S∞ gives a geometric construction of classifying space
s for certain groups that act freely it, such as the cyclic group with two elements and the circle group. The unitary group U in Bott's sense has a classifying space BU for complex vector bundle
s (see Classifying space for U(n)
). A deeper application coming from Kuiper's theorem is the proof of the Atiyah–Jänich theorem (after Klaus Jänich and Michael Atiyah
), stating that the space of Fredholm operator
s on H, with the norm topology, represents the functor K(.) of topological (complex) K-theory, in the sense of homotopy theory. This is given by Atiyah.
of infinite dimension. Here there are only partial results. Some classical sequence spaces have the same property, namely that the group of invertible operators is contractible. On the other hand, there are examples known where it fails to be a connected space
. Where all homotopy groups are known to be trivial, the contractibility in some cases may remain unknown.
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...
, Kuiper's theorem (after Nicolaas Kuiper
Nicolaas Kuiper
Nicolaas Hendrik "Nico" Kuiper was a Dutch mathematician, known for Kuiper's test and proving Kuiper's theorem. He also contributed to the Nash embedding theorem.Kuiper completed his Ph.D...
) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H. It states that the space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
GL(H) of invertible bounded
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
endomorphisms H is such that all maps from any finite complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
Y to GL(H) are homotopic to a constant, for the norm topology on operators.
A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible
Weakly contractible
In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.-Property:It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible.-Example:...
, ie. all its homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s are trivial. This result has important uses in topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
.
General topology of the general linear group
For finite dimensional H, this group would be a complex general linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
and not at all contractible. In fact it is homotopy equivalent to its maximal compact subgroup
Maximal compact subgroup
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups....
, the unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
U of H. The proof that the complex general linear group and unitary group have the same homotopy type is by the Gram-Schmidt process, or through the matrix polar decomposition, and carries over to the infinite-dimensional case of separable Hilbert space, basically because the space of upper triangular matrices is contractible as can be seen quite explicitly. The underlying phenomenon is that passing to infinitely many dimensions causes much of the topological complexity of the unitary groups to vanish; but see the section on Bott's unitary group, where the passage to infinity is more constrained, and the resulting group has non-trivial homotopy groups.
Historical context and topology of spheres
It is a surprising fact that the unit sphereUnit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
, sometimes denoted S∞, in infinite-dimensional Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H is a contractible space
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point....
, while no finite-dimensional spheres are contractible. This result, certainly known decades before Kuiper's, may have the status of mathematical folklore, but it is quite often cited. In fact more is true: S∞ is diffeomorphic to H, which is certainly contractible by its convexity. One consequence is that there are smooth counterexamples to an extension of the Brouwer fixed-point theorem to the unit ball in H. The existence of such counter-examples that are homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s was shown in 1943 by Shizuo Kakutani
Shizuo Kakutani
was a Japanese-born American mathematician, best known for his eponymous fixed-point theorem.Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. Early in his career he spent two years at the Institute for Advanced Study in Princeton at the invitation of the...
, who may have first written down a proof of the contractibility of the unit sphere. But the result was anyway essentially known (in 1935 Andrey Nikolayevich Tychonoff
Andrey Nikolayevich Tychonoff
Andrey Nikolayevich Tikhonov was a Soviet and Russian mathematician known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also inventor of magnetotellurics method in geology. Tikhonov originally published in German, whence the...
showed that the unit sphere was a retract of the unit ball).
The result on the group of bounded operators was proved by the Dutch mathematician Nicolaas Kuiper
Nicolaas Kuiper
Nicolaas Hendrik "Nico" Kuiper was a Dutch mathematician, known for Kuiper's test and proving Kuiper's theorem. He also contributed to the Nash embedding theorem.Kuiper completed his Ph.D...
, for the case of a separable Hilbert space; the restriction of separability was later lifted. The same result, but for the strong operator topology
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
rather than the norm topology, was published in 1963 by Jacques Dixmier
Jacques Dixmier
Jacques Dixmier is a French mathematician. He worked on operator algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace. He received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes.-Publications:*J. Dixmier,...
and Adrien Douady
Adrien Douady
Adrien Douady was a French mathematician.He was a student of Henri Cartan at the Ecole Normale Supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex analytic spaces...
. The geometric relationship of the sphere and group of operators is that the unit sphere is a 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,...
for the unitary group U. The stabiliser of a single vector v of the unit sphere is the unitary group of the orthogonal complement of v; therefore the homotopy long exact sequence predicts that all the homotopy groups of the unit sphere will be trivial. This shows the close topological relationship, but is not in itself quite enough, since the inclusion of a point will be a weak homotopy equivalence only, and that implies contractibility directly only for a CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
. In a paper published two years after Kuiper's, Richard Palais provided technical results on infinite-dimensional manifolds sufficient to resolve this issue.
Bott's unitary group
There is another infinite-dimensional unitary group, of major significance in homotopy theory, that to which the Bott periodicity theoremBott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy...
applies. It is certainly not contractible. The difference from Kuiper's group can be explained: Bott's group is the subgroup in which a given operator acts non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis {ei}, for some N, being the identity on the remaining basis vectors.
Applications
An immediate consequence, given the general theory of fibre bundles, is that every Hilbert bundle is a trivial bundle.The result on the contractibility of S∞ gives a geometric construction of classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
s for certain groups that act freely it, such as the cyclic group with two elements and the circle group. The unitary group U in Bott's sense has a classifying space BU for complex vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s (see Classifying space for U(n)
Classifying space for U(n)
In mathematics, the classifying space for the unitary group U is a space B together with a universal bundle E such that any hermitian bundle on a paracompact space X is the pull-back of E by a map X → B unique up to homotopy.This space with its universal fibration may be constructed as either# the...
). A deeper application coming from Kuiper's theorem is the proof of the Atiyah–Jänich theorem (after Klaus Jänich and Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
), stating that the space of Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
s on H, with the norm topology, represents the functor K(.) of topological (complex) K-theory, in the sense of homotopy theory. This is given by Atiyah.
Case of Banach spaces
The same question may be posed about invertible operators on any Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
of infinite dimension. Here there are only partial results. Some classical sequence spaces have the same property, namely that the group of invertible operators is contractible. On the other hand, there are examples known where it fails to be a connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. Where all homotopy groups are known to be trivial, the contractibility in some cases may remain unknown.