The topology of smooth projective planes
type: Item
instance of: scholarly article
title: The topology of smooth projective planes
zbMATH Open document ID: 0831.51011
DOI: 10.1007/BF01196303
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: 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.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51H25
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 57N15
zbMATH DE Number: 687090
zbMATH Keywords: smooth projective plane
zbMATH Keywords: Hirzebruch signature theorem
zbMATH Keywords: topological \(m\)- plane bundle
zbMATH Keywords: Prontryagin class
zbMATH Keywords: Euler class
zbMATH Keywords: real division algebras
author: Linus Kramer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 687090