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, 19-30 (1964)

• 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