nLab
Kuiper's theorem

Statement

Theorem

(Kuiper’s theorem)

For $\mathscr{H}$ a separable infinite-dimensional complex Hilbert space, the group of bounded and invertible linear operators $\mathrm{GL}(\mathscr{H})$, regarded as a topological group under the norm topology or strong operator topology or weak operator topology, is contractible.

The unitary group $U(\mathscr{H})$, being homotopy equivalent to $\mathrm{GL}(\mathscr{H})$ by the Gram-Schmidt process, is also contractible.

The original paper of Kuiper proved this group to be contractible in the norm topology; later Dixmier and Douady proved contractibility for the strong operator topology. Atiyah and Segal note in their paper on twisted K-theory that there is an easy proof of contractibility in the weak operator topology. One major difference in the topologies is that with the operator topology then it is a CW-complex but with the weak topology then it isn’t even an ANR.

References

• Wikipedia article on Kuiper’s theorem