nLab Kuiper's theorem

Statement

Throughout, let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space.

Theorem

(Kuiper’s theorem)
The topological unitary group $\mathrm{U}(\mathcal{H})$ in either the

• norm topology

or the equivalent

• strong operator topology

• weak operator topology

• compact-open topology

is contractible in that there is a left homotopy between the identity $id \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$ and the constant function $const_{\mathrm{e}} \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$.

Similarly, since the general linear group $GL(\mathcal{H})$ of bounded operators is homotopy-equivalent to $\mathrm{U}(\mathcal{H})$ by the Gram-Schmidt process, it, too, is contractible.

See the commented list of references below.

Related statements

• the infinite-dimensional sphere is weakly contractible

• $U(\mathcal{H})$

References

Proof for the norm topology on U(ℋ):

• Nicolaas Kuiper: Contractibility of the unitary group in Hilbert space, Topology 3 (1964) 19-30

• Luc Illusie: Contractibilité du groupe linéaire des espaces de Hilbert de dimension infinie, Séminaire Bourbaki: années 1964/65 1965/66, exposés 277-312, Séminaire Bourbaki, no. 9 (1966), Exposé no. 284, 9 p. [numdam:SB_1964-1966__9__105_0]

Proof for the strong operator topology on U(ℋ):

• Jacques Dixmier, Adrien Douady, Lem. 3 on p. 251 of: Champs continus d’espaces hilbertiens et de $C^\ast$-algèbres, Bulletin de la Société Mathématique de France, Tome 91 (1963), pp. 227-284 [numdam:BSMF_1963__91__227_0]

Direct proof for the compact-open topology on U(ℋ):

• Michael Atiyah, Graeme Segal, Prop. A2.1 in: Twisted K-theory, Ukrainian Math. Bull. 1, 3 (2004) [arXiv:math/0407054, journal page, published pdf]

Proof that the compact-open topology agrees with the strong operator topology on U(ℋ), by which contractibility of the former follows from that of the latter:

• Martin Schottenloher, The Unitary Group In Its Strong Topology (arXiv:1309.5891), Advances in Pure Mathematics 08 05 (2018) (doi:10.4236/apm.2018.85029)

• Jesus Espinoza, Bernardo Uribe, Topological properties of the unitary group, JP Journal of Geometry and Topology 16 (2014) Issue 1, pp 45-55 (arXiv:1407.1869, journal)

See also:

• Wikipedia, Kuiper’s theorem

Generalization from Hilbert spaces to Hilbert modules:

• Joachim Cuntz, Nigel Higson: Kuiper’s Theorem for Hilbert Modules, in Operator Algebras and Mathematical Physic Contemporary Mathematics 62, American Mathematical Society (1987) 429-435 [ISBN:978-0-8218-7652-7, pdf]

which implies notably the equivariant version of Kuiper’s theorem:

• Heath Emerson, Ralf Meyer, Cor. 3.12 in: Equivariant representable K-theory, Journal of Topology 2 1 (2009) 123-156 [arXiv:0710.1410, doi:10.1112/jtopol/jtp003]