The point and line space of a compact projective plane are homeomorphic (Q1568200)

The author uses the Kister-Mazur Theorem on microbundles together with information on the homotopy groups of classifying spaces for various types of bundles in order to prove the following result. Let \(\xi\) and \(\xi'\) be orientable topological \(\mathbb{R}^n\)-bundles over an \(n\)-dimensional CW-complex. If \(\xi\) and \(\xi'\) are fibre homotopy equivalent and stably equivalent, then they are equivalent. In particular, \(\xi\) is equivalent to the `upside down' bundle obtained by compactifying the fibers and deleting the zero section. This last result has an interesting consequence for compact projective planes in the sense of [\textit{H. Salzmann} et al., Compact Projective Planes, de Gruyter (1996; Zbl 0851.51003)]. Deleting a point \(p\) from the point set of a projective plane, one obtains the total space of an \(\mathbb{R}^n\)-bundle over the line pencil of \(p\); the bundle map is given by joining a point \(q\neq p\) to \(p\). This so-called Hopf bundle comes in two versions dual to each other, i.e., the roles of points and lines are interchanged. On the other hand, geometric considerations show that the two bundles are in the `upside down' relation. This proves the result announced in the title. The geometric considerations mentioned essentially go back to \textit{E. Eisele} [Arch. Math. 58, No. 6, 615-620 (1992; Zbl 0768.51013)], who proved a special case of the result about projective planes.