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