Translation Planes

Flocks of topological circle planes

“…Proof. By [12,Theorem 4.13] the spread S is Pappian if and only if S is a regular spread, i.e., all reguli that contain three lines of S are contained in S. By Theorem 3.1(b) this is equivalent to O :¼ gðSÞ satisfying the following condition: the singular points of a plane spanned by three points of O are contained in O. Now assume this condition and let p 1 and p 2 be two planes of PGðV Þ such that the singular points are contained in O and such that p 1 V p 2 is a line containing two singular points p and q.…”

Quasifields of symplectic translation planes

“…Basic information on spreads, translation planes and quasifields can be found in [4] or [5]. A quasifield is an algebraic structure (Q , +, ·) which satisfies all axioms for a skew field with the possible exception of the associative law for the multiplication and the distributive law x( y + z) = xy + xz.…”

Lie Groups as Multiplication Groups of Topological Loops

“…These quasifields Q coordinatize non-desarguesian 8-dimensional topological translation planes and are determined by N. Knarr ([13], Section 6). Using the results of [13] we have proved that the multiplicative loops Q * are the direct products of R and a compact loop S homeomorphic to S 3 and having the group SL 4 (R) as its multiplication group (cf. [7]).…”