Historical Context and Topology of Spheres
It is a surprising fact that the unit sphere, 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 homeomorphisms 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.
Read more about this topic: Kuiper's Theorem
Famous quotes containing the words historical, context and/or spheres:
“We need a type of theatre which not only releases the feelings, insights and impulses possible within the particular historical field of human relations in which the action takes place, but employs and encourages those thoughts and feelings which help transform the field itself.”
—Bertolt Brecht (1898–1956)
“The hard truth is that what may be acceptable in elite culture may not be acceptable in mass culture, that tastes which pose only innocent ethical issues as the property of a minority become corrupting when they become more established. Taste is context, and the context has changed.”
—Susan Sontag (b. 1933)
“The world has already learned that woman has other virtues than meekness, patience, humility and endurance. She possesses courage above all fear, and a will that knows no obstacles; and when these are called forth by some great emergency, false modesty is trampled in the dust, and spheres are scattered to the winds.”
—A. Holley, U.S. women’s magazine contributor. The Lily, p. 38 (May 1852)