Kuiper's Theorem - Applications

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 spaces 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 bundles (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 operators 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.