Kuiper's Theorem - Historical Context and Topology of Spheres

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.