Flocks in topological circle planes and their representation in generalised quadrangles
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Nils Rosehr
Abstract
In this paper we prove that there is a flock in every locally compact 2-dimensional Laguerre plane, i.e., a set of disjoint circles which covers the point space. We then determine the homeomorphism type of these flocks and use the relation between circle planes and generalised quadrangles to obtain analogous results about generalised quadrangles, Möbius planes and Minkowski planes.
Funding statement: This research was supported by a DAAD fellowship.
Communicated by: R. Löwen
Acknowledgements
I would like to thank the referee for invaluable comments.
References
[1] T. Buchanan, H. Hähl, R. Löwen, Topologische Ovale. Geom. Dedicata 9 (1980), 401–424. MR596738 Zbl 0453.5100710.1007/BF00181558Search in Google Scholar
[2] M. Forst, Topologische 4-Gone. Mitt. Math. Sem. Giessen no. 147 (1981), 65–129. MR628101 Zbl 0478.51013Search in Google Scholar
[3] H. Groh, Topologische Laguerreebenen. I. Abh. Math. Sem. Univ. Hamburg 32 (1968), 216–231. MR234343 Zbl 0165.2360310.1007/BF02993130Search in Google Scholar
[4] H. Groh, Topologische Laguerreebenen. II. Abh. Math. Sem. Univ. Hamburg 34 (1969/70), 11–21. MR257857 Zbl 0187.4230210.1007/BF02992884Search in Google Scholar
[5] W. S. Massey, A basic course in algebraic topology. Springer 1991. MR1095046 Zbl 0725.5500110.1007/978-1-4939-9063-4Search in Google Scholar
[6] S. B. Nadler, Jr., Hyperspaces of sets. Dekker 1978. MR500811 Zbl 0432.54007Search in Google Scholar
[7] N. Rosehr, Flocks of topological circle planes. Dissertation, Würzburg 1999.Search in Google Scholar
[8] N. Rosehr, Flocks of topological circle planes. Adv. Geom. 5 (2005), 559–582. MR2174482 Zbl 1088.5100710.1515/advg.2005.5.4.559Search in Google Scholar
[9] A. E. Schroth, Three-dimensional quadrangles and flat Laguerre planes. Geom. Dedicata 36 (1990), 365–373. MR1076871 Zbl 0717.5100710.1007/BF00150801Search in Google Scholar
[10] A. E. Schroth, The Apollonius problem in flat Laguerre planes. J. Geom. 42 (1991), 141–147. MR1132843 Zbl 0746.5100710.1007/BF01231874Search in Google Scholar
[11] A. E. Schroth, Topological circle planes and topological quadrangles. Longman 1995. MR1381901 Zbl 0839.51013Search in Google Scholar
[12] G. F. Steinke, Topological circle geometries. In: Handbook of incidence geometry, 1325–1354, North-Holland 1995. MR1360739 Zbl 0824.5101210.1016/B978-044488355-1/50026-8Search in Google Scholar
[13] R.-D. Wölk, Topologische Möbiusebenen. Math. Z. 93 (1966), 311–333. MR198332 Zbl 0143.4400110.1007/BF01111942Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
Articles in the same Issue
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
