Results Math 72 (2017), 1021–1030
2017 Springer International Publishing
published online March 31, 2017
Results in Mathematics
Compactness of the Automorphism Group
of a Topological Parallelism on Real
Dieter Betten and Rainer L¨owen
In memoriam Karl Strambach
Abstract. We conjecture that the automorphism group of a topological
parallelism on real projective 3-space is compact. We prove that at least
the identity component of this group is, indeed, compact.
Mathematics Subject Classiﬁcation. 51H10, 51A15, 51M30.
Keywords. Parallelism, automorphism group, compactness.
The notion of a topological parallelism on real projective 3-space PG(3, R)
generalizes the classical example, the Cliﬀord parallelism. Such a parallelism
Π may be considered as a set of spreads such that every line belongs to exactly
one of them and some continuity property holds. Many examples of non-
classical topological parallelisms have been constructed in a series of papers
by Betten and Riesinger, see, e.g.,  and references given therein. The group
Φ = AutΠ of automorphisms of a topological parallelism is a closed subgroup
of PGL(4, R). As is well known, the automorphism group of the Cliﬀord paral-
lelism is the 6-dimensional group PSO(4, R)
SO(3, R)×SO(3, R). Betten and
Riesinger  proved that no other topological parallelism has a group of dimen-
sion dim Φ ≥ 5. Using the result of the present paper, this has been improved
by L¨owen : in fact, the Cliﬀord parallelism is characterized by dim Φ ≥ 4.
Examples of parallelisms with 1-, 2- or 3-dimensional automorphism groups