Locally projective spaces which satisfy the bundle theorem
Let \((M,{\mathcal M})\) be a linear space of dimension \(\geq 3\). For many applications it is interesting to know whether or not \(M\) can be embedded into a projective space (of the same dimension). If \(M\) can be embedded, and each plane in \(M\) satisfies the exchange condition, then the bundle theorem holds. In a famous paper [Math. Z. 175, 219-247 (1980; Zbl 0443.06010)] \textit{J. Kahn} proved for \(\dim M = 3\) that if \(M\) satisfies the bundle theorem, is locally projective, and each plane in \(M\) satisfies the exchange condition, then \(M\) can be embedded into a projective space. The case \(\dim M \geq 4\) has been worked out much earlier, and is easier to handle. Kahn states his theorem for semimodular lattices, but there is no essential difference. The present paper has three aims. Firstly, and most importantly, a considerably simpler and shorter proof of Kahn's theorem is given, which is not restricted to dimension 3. Secondly, the assumption that planes in \(M\) are exchange, is shown to be redundant, i.e., it follows from the other hypotheses. Thirdly, the main tool is the bundle theorem, which is more natural. The use of local projectivity is restricted to a minimum, so parts of the proof can be adapted to spaces which are not locally projective.
scientific article; zbMATH DE number 913037
scientific article