THEOREMS OF POINTS AND PLANES IN THREE-DIMENSIONAL PROJECTIVE SPACE

16 - dimensional compact projective planes with 3 fixed points

Let P 1⁄4 ðP;LÞ be a topological projective plane with a compact point set P of finite (covering) dimension d 1⁄4 dimP > 0. A systematic treatment of such planes can be found in the book Compact Projective Planes [15]. Each line L A L is homotopy equivalent to a sphere Sl with l j 8, and d 1⁄4 2l, see [15] (54.11). In all known examples, L is in fact homeomorphic to Sl. Taken with the compact-o...

Two Coordinatization Theorems for Projective Planes

Since the points of a line uniquely determine the line, it is safe to talk about lines as sets of points, but for convenience we will often regard them separately. We will talk about projective planes using the usual language of points and lines “lies on”, “passes through”, “collinear”, “concurrent”, etc. The third requirement implies its dual, that there are four lines, no three of which are c...

GENERALIZED THEOREMS OF DESARGUES FOR w-DIMENSIONAL PROJECTIVE SPACE

The purpose of the present note is to generalize to n dimensions the celebrated two-triangle theorem of Desargues and its converse.1 The generalized theorems can be stated in simple forms which, nevertheless, suffice for each of the large number of special cases of the configurations involved. The terms concurrent and collinear will be used in the sense that n lines are concurrent if there is a...

Oriented Rank Three Matroids and Projective Planes

Quite recently, GOODMAN, POLLACK, WENGER and ZAMFIRESCU [7], [8] have proven two conjectures by GRÜNBAUM right, showing that any arrangement of pseudolines in the plane can be embedded into a flat projective plane and that there exists a universal topological projective plane in which every arrangement of pseudolines is stretchable. By FOLKMAN and LAWRENCE's theorem [6], this plane contains eve...

Area-minimizing Projective Planes in Three-manifolds