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Kuiper's theorem

Contents

Statement

Theorem

(Kuiper’s theorem)

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

The unitary group U()U(\mathcal{H}), being homotopy equivalent to GL()GL(\mathcal{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

Revised on April 9, 2013 19:23:07 by Urs Schreiber (131.174.41.18)