Compactness of the automorphism group of a topological parallelism on real projective 3-space: the disconnected case (Q668632)

Clifford parallelism on real projective 3-space \(\mathrm{PG}(3,\mathbb{R})\) is the classical example of a topological parallelism. In general, a topological parallelism is an equivalence relation on the set of lines such that every equivalence class \(C\) is a spread (that is, its elements form a simple cover of the point set) and such that some continuity condition is satisfied. The author proves that the automorphism group of a topological parallelism on real projective 3-space is compact. This settles a conjecture stated in [\textit{D. Betten} and \textit{R. Löwen}, Result. Math. 72, No. 1--2, 1021--1030 (2017; Zbl 1379.51003)], where it was proved that at least the connected component of the identity is compact.