The topology of smooth projective planes
Let \(({\mathcal P}, {\mathcal L}, {\mathcal F})\) be a smooth projective plane, i.e. \(\mathcal P\) and \(\mathcal L\) are smooth manifolds and the geometric operations of join and intersection are smooth mappings. The author shows that then \(\mathcal P\) and \(\mathcal L\) are both homeomorphic to the point space of the projective plane over one of the four alternative real division algebras. The question whether these manifolds are even diffeomorphic remains open. Roughly, the proof runs as follows. Assume that \(\dim {\mathcal P}= \dim {\mathcal L}= m\). Let \(o\in {\mathcal P}\) and \(L\in {\mathcal L})\) be non-incident. Then the mapping \(h: {\mathcal P}\smallsetminus\{o\}\to L\) which sends \(p\) to \((p\vee o)\vee L\) is a locally trivial topological \(m\)-plane bundle \(\eta\) over the \(m\)-sphere \(L\), and \(\mathcal P\) is homeomorphic to the Thom space of this bundle. By a construction of the reviewer [Forum Math. 2, No. 6, 603-612 (1990; Zbl 0711.51002)], the flat space \(\mathcal F\) can be embedded as a hyperplane into a sphere of dimension \(3m+ 1\). This allows the computation of the Pontryagin class of \(\mathcal P\) and of the Euler and Pontryagin classes of \(\eta\). Comparison with the classical case then yields the desired result.
scientific article; zbMATH DE number 687090
scientific article