On isomorphisms of Grassmann spaces (Q1291169)

The author constructs an embedding \(\phi:P\to P'\) of a projective space \((P,\mathcal L)\) of dimension greater than \(2\) into a projective plane \((P',\mathcal L')\) such that \(| L\cap\phi(P)|\geq 2\) for all \(L\in\mathcal L'\) (formula~(3) is miss-printed). Therefore \(\phi\) is not surjective, hence the induced map on the Grassmann space maps intersecting lines onto intersecting lines, but not vice versa, and is therefore not an isomorphism. This contradicts Satz~2 in [\textit{H. Brauner}, Akad. Wiss. Math.-Naturwiss. Kl. 197, No. 4-7, 327-332 (1988; Zbl 0668.51002)], and solves an open problem for Grassmann spaces. The construction is done by direct limits over suitably chosen vector spaces.