A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology
Matthew R. Brown
Ghent University Department of Pure Mathematics and Computer Algebra Galglaan 2 Gent B-9000 Belgium
DOI: 10.1023/A:1013812619953
Abstract
If a GQ S O x O_x of S Q Q (4,q),q even, and O x O_x is an elliptic quadric for each X Q Q (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.
Pages: 107–125
Keywords: generalized quadrangle; subquadrangle; cohomology; ovoid
Full Text: PDF
References
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2. M.R. Brown, “Generalized Quadrangles and Associated Structures,” Ph.D. Thesis, University of Adelaide, 1997.
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12. S.E. Payne, “A restriction on the parameters of a subquadrangle,” Bull. Amer. Math. Soc. 70 (1973), 747-748.
13. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Pitman, London, 1984.
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19. J.A. Thas and S.E. Payne, “Spreads and ovoids in finite generalized quadrangles,” Geom. Dedicata 52(3) (1994), 227-253.
2. M.R. Brown, “Generalized Quadrangles and Associated Structures,” Ph.D. Thesis, University of Adelaide, 1997.
3. M.R. Brown, “Semipartial geometries and generalized quadrangles of order (r, r2),” Bull. Belg. Math. Soc. Simon Stevin 5(2/3) (1998), 187-205.
4. P.J. Cameron, “Covers of graphs and EGQs,” Discrete Math. 97(1-3) (1991), 83-92.
5. F. De Clerck and H. Van Maldeghem, “Some classes of rank 2 geometries,” in The Handbook of Incidence Geometry, Elsevier Science Publishers, B.V., Amsterdam, The Netherlands, 1995, pp. 433-475.
6. M.J. Greenberg and J.R. Harper, Algebraic Topology, A First Course, The Benjamin/Cummings Publishing, Reading, MA, 1981.
7. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, 2nd ed., Clarendon Press, Oxford, 1998.
8. J.W.P. Hirschfeld and J.A. Thas, “Sets of type (1, n, q + 1) in PG(d, q),” Proc. London Math. Soc. 41(3) (1980), 254-278.
9. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Clarendon Press, Oxford, 1991.
10. W.M. Kantor, “Ovoids and translation planes,” Canad. J. Math. 34 (1980), 1195-1203.
11. W.M. Kantor, “Some generalized quadrangles with parameters q2, q,” Math. Z. 192 (1986), 45-50.
12. S.E. Payne, “A restriction on the parameters of a subquadrangle,” Bull. Amer. Math. Soc. 70 (1973), 747-748.
13. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Pitman, London, 1984.
14. J.J. Seidel, “A survey of two-graphs,” in Accad. Naz. Lincei. Roma, 1976, pp. 481-511.
15. H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, New York, 1980.
16. E.H. Spanier, Algebraic Topology, Springer-Verlag, New York, 1966.
17. J.A. Thas, “A remark concerning the restriction on the parameters of a 4-gonal configuration,” Simon Stevin 48 (1974/75), 65-68.
18. J.A. Thas, “3-regularity in generalized quadrangles: A survey, recent results and the solution of a longstanding conjecture,” Rend. Circ. Mat. Palermo 2 (Suppl. 53) (1998), 199-218. Combinatorics '98 (Mondello).
19. J.A. Thas and S.E. Payne, “Spreads and ovoids in finite generalized quadrangles,” Geom. Dedicata 52(3) (1994), 227-253.