Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Flag-homogeneous compact connected polygons

  • 25 Accesses

  • 21 Citations

Abstract

The flag-homogeneous compact connected polygons with equal topological parametersp = q are classified explicitly. These polygons turn out to be Moufang polygons.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Adams, J. F. and Atiyah, M. F.:K-theory and the Hopf invariant,Quart. J. Math. Oxford (2) 17 (1966), 31–38.

  2. 2.

    Arens, R.: Topologies for homeomorphism groups,Amer. J. Math. 68 (1946), 593–610.

  3. 3.

    Ball, B.J.: Arcwise connectedness and the persistence of errors,Amer. Math. Monthly 91 (1984), 431–433.

  4. 4.

    Bickel, H.: Lie-projektive Gruppen: Vergeleich von Approximationsbegriffen, Diplomarbeit Braunschweig, 1992.

  5. 5.

    Borel, A.: La cohomologie mod 2 de certains espaces homogénes,Comment. Math. Helv. 27 (1953), 165–197.

  6. 6.

    Borel, A. and De Siebenthal, J.: Les sous-groupes fermés de rang maximum des groupes de Lie clos,Comment. Math. Helv. 23 (1949), 200–221.

  7. 7.

    Brouwer, A. E.: A non-degenerate generalized quadrangle with lines of size four is finite, in J. W. P. Hirschfeldet al. (eds)Proc. Brighton 1990, Oxford University Press 1991, pp. 47–49.

  8. 8.

    Burns, K. and Spatzier, R.: On topological Tits buildings and their classification,Publ. Math. I.H.E.S. 65 (1987), 5–34.

  9. 9.

    Dienst, K. J.: Verallgemeinerte Vierecke in projektiven Räumen,Arch. Math. 35 (1980), 177–186.

  10. 10.

    Dugundji, J.:Topology, Allyn and Bacon, Boston, 1966.

  11. 11.

    Engelking, R.:General Topology, Heldermann Verlag, Berlin, 1989.

  12. 12.

    Engelking, R. and Sieklucki, K.:Topology. A Geometric Approach, Heldermann Verlag, Berlin, 1992.

  13. 13.

    Feit, W.: Finite projective planes and a question about primes,Proc. Amer. Math. Soc. 108 (1990), 561–564.

  14. 14.

    Ferus, D., Karcher, H. and Münzner, H.-F.: Cliffordalgebren und neue isoparametrische Hyperflächen,Math. Z. 177 (1981), 479–502.

  15. 15.

    Fink, J. B.: Flag-transitive projective planes,Geom. Dedicata 17 (1985), 219–226.

  16. 16.

    Freudenthal, H.: Einige Sätze über topologische Gruppen,Ann. Math. 37 (1936), 44–56.

  17. 17.

    Gluškov, V. M.: The structure of locally compact groups and Hilbert's fifth problem,Amer. Math. Soc. Transl. (2),15 (1960), 55–93.

  18. 18.

    Grove, K. and Halperin, S.: Dupin hypersurfaces, group actions, and the double mapping cylinder,J. Differential Geom. 26 (1987), 429–459.

  19. 19.

    Grundhöfer, T.: Ternary fields of compact projective planes,Abh. Math. Sem. Univ. Hamburg 57 (1986), 87–101.

  20. 20.

    Grundhöfer, T.: Automorphism groups of compact projective planes,Geom. Dedicata 21 (1986), 291–298.

  21. 21.

    Grundhöfer, T. and Knarr, N.: Topology in generalized quadrangles,Topology Appl. 34 (1990), 139–152.

  22. 22.

    Grundhöfer, T. and Van Maldeghem, H.: Topological polygons and affine buildings of rank three,Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 459–479.

  23. 23.

    Hähl, H.: Automorphismengruppen achtdimensionaler lokalkompakter Quasikörper,Math. Z. 149 (1976), 203–225.

  24. 24.

    Hebda, J. J.: The possible cohomology rings of certain types of taut submanifolds,Nagoya Math. J. 111 (1988), 85–97.

  25. 25.

    Helgason, S.:Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, San Diego, 1978.

  26. 26.

    Hsiang, W.-Y. and Lawson, H. B.: Minimal submanifolds of low cohomogeneity,J. Differential Geom. 5 (1971), 1–38.

  27. 27.

    Hothi, A.: Another alternative proof of Effros' Theorem,Topology Proc. 12 (1987), 256–298.

  28. 28.

    Kantor, W. M.: Flag-transitive planes, in C. A. Baker and L. M. Batten (eds.),Finite Geometries, Proc. Winnipeg 1984, Dekker, New York, 1985, pp. 179–181.

  29. 29.

    Kantor, W. M.: Primitive permutation groups of odd degree, and an application to finite projective planes,J. Algebra 106 (1987), 15–45.

  30. 30.

    Knarr, N.: Projectivities of generalized polygons,Ars Combin. 25B (1988), 265–275.

  31. 31.

    Knarr, N.: The nonexistence of certain topological polygons,Forum Math. 2 (1990), 603–612.

  32. 32.

    Kramer, L.: Gebäude auf isoparametrischen Untermannigfaltigkeiten, Diplomarbeit Tübingen, 1991.

  33. 33.

    Kuratowski, K.:Topology, Vol. II, Academic Press, New York, 1968.

  34. 34.

    Löwen, R.: Homogeneous compact projective planes,J. reine angew. Math. 321 (1981), 217–220.

  35. 35.

    Lunardon, G. and Pasini, A.: FiniteC n geometries: a survey,Note di Matematica 10 (1990), 1–35.

  36. 36.

    Montgomery, D.: Simply connected homogeneous spaces,Proc. Amer. Math. Soc. 1 (1950), 467–469.

  37. 37.

    Montgomery, D. and Zippin, L.:Topological Transformation Groups, Interscience Publishers, New York, 1955.

  38. 38.

    Münzner, H.-F.: Isoparametrische Hyperflächen in Sphären II,Math. Ann. 256 (1981), 215–232.

  39. 39.

    Ronan, M.:Lectures on Buildings, Academic Press, San Diego, 1989.

  40. 40.

    Salzmann, H.: Topologische projektive Ebenen,Math. Z. 67 (1957), 436–466.

  41. 41.

    Salzmann, H.: Kompakte zweidimensionale projektive Ebenen,Math. Ann. 145 (1962), 401–428.

  42. 42.

    Schurle, A. W.:Topics in Topology, Elsevier North Holland, New York, 1979.

  43. 43.

    Szenthe, J.: On the topological characterization of transitive Lie group actions,Acta Sci. Math. (Szeged) 36 (1974), 323–344.

  44. 44.

    Takagi, R. and Takahashi, T.: On the principal curvatures of homogeneous hypersurfaces in a sphere, in S. Kobayashi, M. Obata and T. Takahashi (eds),Differential Geometry, in honor of Kentaro Yano, Kinokuniya, Tokyo, 1972, pp. 469–481.

  45. 45.

    Thorbergsson, G.: Clifford algebras and polar planes,Duke Math. J. 67 (1992), 627–632.

  46. 46.

    Tits, J.: Sur la trialité et certaines groupes qui s'en déduisent,Publ. Math. I.H.E.S. 2 (1959), 14–60.

  47. 47.

    Tits, J.:Buildings of Spherical Type and Finite BN-Pairs (2nd edn), Lecture Notes in Maths 386, Springer, Berlin, 1986.

  48. 48.

    Wang, H.-C.: Homogeneous spaces with non-vanishing Euler characteristics,Ann. Math. (2) 50 (1949), 469–481.

  49. 49.

    Wang, Q.-M.: On the topology of Clifford isoparametric hypersurfaces,J. Diff. Geom. 27 (1988), 55–66.

  50. 50.

    Warner, G.:Harmonic Analysis on Semi-simple Lie Groups, Springer, 1972.

  51. 51.

    Wolf, J. A.:Spaces of Constant Curvature, Publish or Perish, Wilmington, 1984.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Grundhöfer, T., Knarr, N. & Kramer, L. Flag-homogeneous compact connected polygons. Geom Dedicata 55, 95–114 (1995). https://doi.org/10.1007/BF02179088

Download citation

Mathematics Subject Classifications (1991)

  • 51H10
  • 51E12
  • 51A10