Eine Kennzeichnung der Oktavenebene. (A characterization of Cayley planes). (German) Zbl 0623.51011

Hähl, Hermann

Indag. Math. 49, 29-39 (1987).

The paper concerns partitions P of \({\mathbb{R}}^{16}\) into 8-dimensional subspaces (i.e. every vector \(\neq 0\) belongs to precisely one element of P). The classical example is the Hopf partition P(\({\mathbb{O}})\) consisting of the lines through the origin of the Cayley plane. The main result is that this is the only partition that is invariant under a linear action of a group locally isomorphic to \(S0_ 7({\mathbb{R}},1)\). Via intersection, partitions correspond to disjoint decompositions of the sphere \(S_{15}\) into great 7-spheres.
The result is connected to the theory of topological translation planes. The translates of a partition P form an affine translation plane A(P), and A(P) is a topological plane if and only if P is closed in the Graßmann manifold \(G_{16,8}\) (also if and only if the corresponding decomposition of \(S_{15}\) is a locally trivial fibration). Non-closed partitions can be obtained by transfinite induction. In contrast to \(S0_ 7({\mathbb{R}},1)\), the compact group \(S0_ 7({\mathbb{R}})\) does act on nonclassical closed partitions [the author, Arch. Math. 48, 267-276 (1987)]. The characterization of P(\({\mathbb{O}})\) given here will be an important step in the author’s classification of 16-dimensional topological translation planes with large collineation groups, cf. Math. Z. 159, 259-294 (1978; Zbl 0381.51009).

Reviewer: R.Löwen

MSC:

51H10	Topological linear incidence structures
55R05	Fiber spaces in algebraic topology

Keywords:

Hopf partition; Cayley plane; topological translation planes