Abstract: By a theorem of Dixmier-Douady, the unitary group of an infinite-dimensional Hilbert space $H$ in the strong operator topology is contractible. The Dixmier-Douady proof is based on an explicit construction of families of subspaces and operators in $H$ with rather special properties. Unfortunately, their proof leaves hidden the geometric meaning of the theorem. In my talk, I will recall the proof of Dixmier-Douady and present another, more geometric proof of this theorem. The talk is based on a recent preprint (arXiv:2504.11646; joint with N. V. Ivanov).