Abstract
It will be proved that the compact connected topological generalized quadrangles which admit a collineation group that acts transitively on ordered pentagons are precisely the real or complex orthogonal quadrangles, up to duality.
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References
Bott, R.: An application of the Morse theory to the topology of Lie groups,Bull. Soc. Math. France 84 (1956), 251–281.
Bourbaki, N.:Algèbre IX, Hermann, Paris, 1973.
Bourbaki, N.:Lie Groups and Lie Algebras, Chapters 1–3, Springer, Berlin, Heidelberg, New York, 1989.
Buekenhout, F. and Van Maldeghem, H.: Finite distance-transitive generalized polygons,Geom. Dedicata 52 (1994), 41–51.
Burns, K. and Spatzier, R.: On topological Tits buildings and their classification,Publ. Math. IHES 65 (1987), 5–34.
De Smet, V. and Van Maldeghem, H.: The finite Moufang hexagons coordinatized,Beiträge Alg. Geom. 34 (1993), 217–232.
Dieudonné, J.:La Géometrie des Groupes Classiques, Springer, Berlin, Heidelberg, New York, 1971.
Grundhöfer, T. and Knarr, N.: Topology in generalized quadrangles,Top. Appl. 34 (1990), 139–152.
Grundhöfer, T., Knarr, N. and Kramer, L.: Flag-homogeneous compact connected polygons,Geom. Dedicata 55(1) (1995), 95–114.
Hanssens, G. and Van Maldeghem, H.: Algebraic properties of quadratic quaternary rings,Geom. Dedicata 30 (1989), 43–67.
Hsiang, W.-Y. and Lawson Jr, H. B.: Minimal submanifolds of low cohomogeneity,J. Differential Geom. 5 (1971), 1–38.
Hughes, D. R. and Piper, F. C.:Projective Planes, Springer, Berlin, Heidelberg, New York, 1973.
Iwasawa, K.: On some types of topological groups,Ann. Math. 50 (1949), 507–558.
Knarr, N.: The nonexistence of certain topological polygons,Forum Math. 2 (1990), 603–612.
Kramer, L.: Compact polygons, Dissertation, Universität Tübingen, 1994.
Löwen, R.: Homogeneous compact projective planes,J. reine angew. Math. 321 (1981), 217–220.
Pickert, G.:Projektive Ebenen, Springer, Berlin, Göttingen, Heidelberg, 1955.
Ronan, M. A.:Lectures on Buildings, Academic Press, San Diego, 1989.
Rotman, J. J.:An Introduction to Algebraic Topology, Springer, Berlin, Heidelberg, New York, 1988.
Salzmann, H.: Homogene kompakte projektive Ebenen,Pacific J. Math. 60, No. 2 (1975), 217–234.
Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R. and Stroppel, M.:Compact Projective Planes, De Gruyter, Berlin, New York, 1995.
Thas, J. A. and Van Maldeghem, H.: The classification of finite generalized quadrangles admitting a group acting transitively on ordered pentagons (to appear).
Tits, J.:Tabellen zu den einfachen Liegruppen und ihren Darstellungen, Springer, Berlin, Heidelberg, New York, 1967.
Tits, J.:Buildings of Spherical Type and Finite BN-Pairs, Springer, Berlin, Heidelberg, New York, 1974.
Van Maldeghem, H.: A configurational characterization of the Moufang generalized polygons,Europ. J. Combin. 11 (1990), 381–392.
Warner, G.:Harmonic Analysis of Semi-simple Lie Groups I, Springer, Berlin, Heidelberg, New York, 1972.
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Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday