We prove that compact connected w-gons in which all point rows and all line pencils are topological manifolds exist only for n = 3, 4 or 6. There are also some restrictions on the possible dimensions of the point rows and the line pencils. 1980 Mathematics Subject Classification (1985 Revision): 51H15, 51H20. Shortly after the introduction of generalized polygons by Tits in 1959, Feit and Higman [8] were able to prove that finite generalized «-gons exist only for n = 3,4,6 or 8. A similar result for (infinite) Moufang w-gons was obtained by Tits [23] and independently by Weiss [24]. Here we show that the Situation for topological polygons with reasonably good topologies is quite analogous; namely we prove the following Theorem. Lei ä? = (P, L, F) be a compact connected n-gon, n > 3. Assume that all point rows and all line pencils are topological manifolds. Then n is equal to3,4or6. The point rows and line pencils are spheres ofdimension p and q, respectively, and there are the following restrictions on p and q: Ifn = 3 or 6 we havep = q = l, 2, 4 or 8.Ifn = 4 and p, q > l, then either p + q is odd or p = q is even.It should be noted that, in contrast to the case of finite or Moufang polygons, n = 8 is not possible in our context. This is no surprise since the existence of generalized octagons seems to be a phenomenon associated with characteristic 2.The proof of the theorem is achieved by constructing a "topological Veronese embedding" of 9 into some sphere. This embedding leads to a Situation which was
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