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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Kuiper's theorem (after
Nicolaas Kuiper Nicolaas Hendrik Kuiper (; 28 June 1920 – 12 December 1994) was a Dutch mathematician, known for Kuiper's test and proving Kuiper's theorem. He also contributed to the Nash embedding theorem. Kuiper studied at University of Leiden in 1937- ...
) is a result on the topology of operators on an infinite-dimensional, complex
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
 ''H''. It states that the
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
GL(''H'') of
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
bounded
endomorphisms In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a grou ...
of ''H'' is such that all maps from any finite complex ''Y'' to GL(''H'') are
homotopic In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
to a constant, for the
norm topology In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informal ...
on operators. A significant corollary, also referred to as ''Kuiper's theorem'', is that this group is weakly contractible, ''ie.'' all its
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s are trivial. This result has important uses in
topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
.


General topology of the general linear group

For finite dimensional ''H'', this group would be a complex
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
and not at all contractible. In fact it is homotopy equivalent to its
maximal compact subgroup In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
, the
unitary group Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semi ...
''U'' of ''H''. The proof that the complex general linear group and unitary group have the same
homotopy type In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
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 In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are z ...
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 sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
, sometimes denoted ''S'', in infinite-dimensional
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
''H'' is a
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 within t ...
, while no finite-dimensional spheres are contractible. This result, certainly known decades before Kuiper's, may have the status of
mathematical folklore In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics, or ...
, but it is quite often cited. In fact more is true: ''S'' is
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Defini ...
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 Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the ...
to the unit ball in ''H''. The existence of such counter-examples that are
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
s was shown in 1943 by
Shizuo Kakutani was a Japanese and American mathematician, best known for his eponymous fixed-point theorem. Biography Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institu ...
, 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 Tikhonov (; 17 October 1906 – 7 October 1993) was a leading USSR, Soviet Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed prob ...
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 Hendrik Kuiper (; 28 June 1920 – 12 December 1994) was a Dutch mathematician, known for Kuiper's test and proving Kuiper's theorem. He also contributed to the Nash embedding theorem. Kuiper studied at University of Leiden in 1937- ...
, for the case of a separable Hilbert space; the restriction of separability was later lifted. The same result, but for the
strong operator topology In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
rather than the norm topology, was published in 1963 by
Jacques Dixmier Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping. Biogra ...
and
Adrien Douady Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician born in La Tronche, Isère. He was the son of Daniel Douady and Guilhen Douady. Douady was a student of Henri Cartan at the École normale supérieure, and initi ...
. The geometric relationship of the sphere and group of operators is that the unit sphere is a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
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 In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with cla ...
only, and that implies contractibility directly only for a
CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
. In a paper published two years after Kuiper's,


Bott's unitary group

There is another infinite-dimensional unitary group, of major significance in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
, that to which the
Bott 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 comple ...
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 , for some ''N'', being the identity on the remaining basis vectors.


Applications

An immediate consequence, given the general theory of
fibre bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s, is that every Hilbert bundle is a
trivial bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. The result on the contractibility of ''S'' gives a geometric construction of
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'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
s for certain groups that act freely on it, such as the cyclic group with two elements and the
circle group In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...
. The unitary group ''U'' in Bott's sense has a classifying space ''BU'' for complex
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 (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s (see
Classifying space for U(n) In mathematics, the classifying space for the unitary group U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a paracompact space ''X'' is the pull-back of EU(''n'') by a map ''X'' → BU('' ...
). A deeper application coming from Kuiper's theorem is the proof of the Atiyah–Jänich theorem (after Klaus Jänich and
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 the ...
), stating that the space of
Fredholm operator In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' :  ...
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 space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
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 In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
.Herbert Schröder
''On the topology of the group of invertible elements'' (PDF), preprint survey
Where all homotopy groups are known to be trivial, the contractibility in some cases may remain unknown.


References

* {{cite journal, last=Kuiper , first=N. , title=The homotopy type of the unitary group of Hilbert space , journal=
Topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, volume=3 , year=1965, issue=1 , pages=19–30 , doi=10.1016/0040-9383(65)90067-4, doi-access= K-theory Operator theory Hilbert spaces Theorems in topology Topology of Lie groups