Compact connected topological planes of dimension d ~< 4 have been studied extensively, and rather conclusive results have been obtained. In particular, the point set P is homeomorphic to the real or complex projective plane, and the lines are one-or two-spheres respectively. Moreover, non-degenerate closed subplanes are always connected. In the compact-open topology, the full collineation group F is a Lie group of dimensionf ~ 4d, containing the automorphism group Z (the group of continuous collineations) as a separable open subgroup. If dim I ~ > 2d, or if E is point-transitive, the plane is isomorphic to the arguesian real or complex plane. Forf = 2d, however, there exist continuously many different non-arguesian planes all of which are known explicitly: The planes with d = 2, f--4 are exactly the Moulton planes [25]. Each plane satisfying d = 4 andf = 8 is a translation plane or the dual thereof [31; 32; 3]. These translation planes have been determined completely by Betten; there are two singular ones in addition to a oneparameter family [2; 4; 5].
Turning now to compact projective planes of dimension d = dim P > 4, the situation is more difficult. At present, there seems to be no way of proving that the lines of such a plane are necessarily manifolds. It is also conceivable that there exist 0-dimensional closed subplanes. The following has been proved however: (a) If a line L is a manifold, it is homeomorphic to the f-sphere 5t with ~ = 4 or 8, and d = 2# [26, 7.12]. (b) In the general case, each line L has trivial homotopy groups rq(L) = rr2(L) = 0 [33]; moreover, the complement of a point in L is locally and globally contractible (by homeomorphisms) and arcwise connected [26, 7.11 ]. (c) The automorphism group Y. is always locally compact and has a countable basis [33, (*)]. (d) If d ~ 16 andZ is transitive on the point set P, or ifE is transitive on the set of incident point-line pairs, the plane is the (real) quaternion or octave plane [33, (I, II)]. The same conclusion holds if the stabilizer Z a of some point a is transitive on Pl{a} [dual of 35]. (e) If ~ is a proper translation plane, and d = 8, then dim E = f ~< 18, and f = 18 implies that the dual of ~ is also a translation plane. All 8-dimensional translation planes with f >t 17 have been determined explicitly by Hiihl 03; 12].