algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
(Kuiper’s theorem)
For $\mathcal{H}$ a separable infinite-dimensional complex Hilbert space, the group of bounded and invertible linear operators $GL(\mathcal{H})$, regarded as a topological group under the norm topology or strong operator topology or weak operator topology, is contractible.
The unitary group $U(\mathcal{H})$, being homotopy equivalent to $GL(\mathcal{H})$ 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.
Note that on $U(\mathcal{H})$ the strong operator topology coincides with the compact open topology (Schottenloher), and with these topologies $U(\mathcal{H})$ is a topological group (the same is not true for $GL(\mathcal{H})$). In fact more is true: the compact open, strong and weak topologies and their $\ast$-counterparts all agree on $U(\mathcal{H})$, which in this topology is a Polish group (Espinoza-Uribe).
Martin Schottenloher, The Unitary Group In Its Strong Topology, web
Jesus Espinoza, Bernardo Uribe, Topological properties of the unitary group, JP Journal of Geometry and Topology 16 (2014) Issue 1, pp 45-55. journal, arXiv:1407.1869