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Topological planes. (English) Zbl 0153.21601
A topological plane is a plane in the sense of an incidence geometry with points and lines, and with a topology in which the operations of joining and intersecting are continuous. Though, of course, there are higher dimensional topological planes, those based on 2-dimensional surfaces (called flat) have been mostly studied. The author of the present survey has himself much contributed to this field. An important means of classification of flat topological planes is their collineation group, in particular because collineation group of a flat plane bears a Lie structure. So it is at most 6-dimensional for affine planes. A plane is called flexible, if a certain incidence pair can be carried into any other in a neighborhood by collineation. The collineation group of a flexible affine plane is at least 3-dimensional. Curiously there is one type of affine plane with a 3-dimensional collineation group which is not flexible, to wit the strip between two parallel lines in the ordinary plane. If a flat plane admits a 6-dimensional collineation group, it is the ordinary affine plane. It is even the only flat plane admitting a 5-dimensional group of collineations. Again the only doubly homogeneous affine flat plane is the ordinary one. 4-dimensional collineation groups can fully be characterized; their planes, if provided with a fixed point, are Moulton planes. Other results are related to such planes with a 1-dimensional orbit. Analogous results follow for projective flat planes, e.g. one with a 4-dimensional collineation group is Moulton or aguesian. This finally leads to the classification of all projective and affine flat planes with a 3-dimensional collineation group.
To tackle higher dimensional planes coordinization by ternary fields is used. By the analysis of locally Euclidean ternary fields it comes out that the lines of projective planes are spheres of dimensions $$1,2,4,8$$ as soon as they are manifolds.
The survey is very well written. it is a pity that it is nearly impossible to trace definitions.

##### MSC:
 51H10 Topological linear incidence structures 51H05 General theory of topological geometry 51-02 Research exposition (monographs, survey articles) pertaining to geometry 51E15 Finite affine and projective planes (geometric aspects)
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##### References:
 [1] Adams, J.F., On the nonexistence of elements of Hopf invariant one, Bull. amer. math. soc., Ann. math., 72, 20-104, (1960) · Zbl 0096.17404 [2] Alexandroff, P.; Hopf, H., Topologie I, (1935), Springer · JFM 61.0602.07 [3] André, J., Über projektive ebenen vom Lenz-Barlotti-typ III2, Math. Z., 84, 316-328, (1964) · Zbl 0119.16201 [4] Baer, R., Homogeneity of projective planes, Amer. J. math., 64, 137-152, (1942) · Zbl 0060.32207 [5] Barlotti, A., Le possibili configurazioni del sistema Della coppie punto-retta (A, a) per cui un piano grafico risulta (A, a)-transitivo, Boll. un. mat. ital., 12, 212-226, (1957) · Zbl 0077.13802 [6] Bing, R.H.; Borsuk, K., Some remarks concerning topological homogeneous spaces, Ann. math., 81, 100-111, (1965) · Zbl 0127.13302 [7] Brouwer, L.E.J., Die theorie der endlichen kontinuierlichen gruppen, unabhängig von den axiomen von Lie, Math. ann., 67, 246-267, (1909) · JFM 40.0194.01 [8] Busemann, H., The geometry of geodesics, (1955), Academic Press Berlin · Zbl 0112.37002 [9] Busemann, H., Metrizations of projective spaces, (), 387-390 · Zbl 0115.39901 [10] Busemann, H., Convex surfaces, (1958), Wiley (Interscience) New York · Zbl 0196.55101 [11] Busemann, H.; Salzmann, H., Metric collineations and inverse problems, Math. Z., 87, 214-240, (1965) · Zbl 0125.11102 [12] Cartan, É., La topologie des groupes de Lie, Paris 1936, Enseignement math., 35, 177-200, (1936), reprinted in · JFM 62.0441.02 [13] Curtis, M.L.; Fort, M.K., Homotopy groups of one-dimensional spaces, (), 140-148 · Zbl 0089.38802 [14] Dieudonné, J., La géométrie des groupes classiques, (1963), Springer New York · Zbl 0111.03102 [15] Dugundji, J., Topology, (1966), Allyn and Bacon Berlin · Zbl 0144.21501 [16] Eilenberg, S., Ordered topological spaces, Amer. J. math., 63, 39-45, (1941) · JFM 67.0756.04 [17] Feigl, G., Fixpunktsätze für spezielle n-dimensionale mannigfaltigkeiten, Math. ann., 98, 355-398, (1928) · JFM 53.0553.04 [18] Freudenthal, H., La structure des groupes à deux bouts et des groupes triplement transitifs, Indag. math., 13, 288-294, (1951) · Zbl 0044.02001 [19] Freudenthal, H., Kompakte projektive ebenen, Illinois J. math., 1, 9-13, (1957) · Zbl 0077.33809 [20] Freudenthal, H., Oktaven, ausnahmegruppen und oktavengeometrie, (1960), Utrecht · Zbl 0100.03006 [21] Freudenthal, H., Lie groups in the foundations of geometry, Advances in math., 1, 145-190, (1964) · Zbl 0125.10003 [22] Gluškov, V.M., On the structure of connected locally bicompact groups, Mat. sb., Amer. math. soc. transl., 27, 2, 159-177, (1963), See also · Zbl 0128.03002 [23] Hall, M., Projective planes, Trans. amer. math. soc., 54, 229-277, (1943) · Zbl 0060.32209 [24] Hall, M., The theory of groups, (1959), Macmillan Boston, Massachusetts [25] Harrold, O.G., Pseudo-isotopically contractible spaces, (), 186-187 · Zbl 0136.43604 [26] Haupt, O., Zur geometrie in topologisch ebenen hyperbolischen ebenen, Bayer. akad. wiss. math.-nat. kl. S.-B., 163-166, (1962) · Zbl 0114.38302 [27] Higman, D.G.; McLaughlin, J.E., Geometric ABA-groups, Illinois J. math., 5, 382-397, (1961) · Zbl 0104.14702 [28] Hilbert, D., Grundlagen der geometrie, (1956), Teubner New York · JFM 56.0481.01 [29] Hofmann, K.H., Topologische doppelloops, Math. Z., 70, 213-230, (1958) · Zbl 0095.02703 [30] Hofmann, K.H., Topologische distributive doppelloops, Math. Z., 71, 36-68, (1959) · Zbl 0095.02704 [31] Hofmann, K.H., Topologische doppelloops und topologische halbgruppen, Math. ann., 138, 239-258, (1959) · Zbl 0095.02801 [32] Hofmann, K.H.; Hofmann, K.H., Lokal kompakte zusammenhängende topologische halbgruppen mit dichter untergruppe, Math. ann., Math. ann., 140, 442-32, (1960) · Zbl 0111.24202 [33] Hofmann, K.H., Über die topologische und algebraische struktur topologischer doppelloops und einiger topologischer projektiver ebenen, (), 57-67 · Zbl 0116.12705 [34] Hofmann, K.H.; Mostert, P., Splitting in topological groups, Mem. amer. math. soc., 43, (1963), 75 pp · Zbl 0163.02705 [35] Hurewicz, W.; Wallman, H., Dimension theory, (1948), Princeton University Press Oxford · JFM 67.1092.03 [36] Jacobson, N., Lie algebras, (1962), Wiley (Interscience) Princeton, New Jersey · JFM 61.1044.02 [37] Kalscheuer, F., Die bestimmung aller stetigen fastkörper, Abh. math. sem. univ. Hamburg, 13, 413-435, (1940) · JFM 66.0106.01 [38] Kerékjártó, B.v., Vorlesungen über topologie I. flächentopologie, (1923), Springer New York · JFM 49.0396.07 [39] Kneser, H., Reguläre kurvenscharen auf den ringflächen, Math. ann., 91, 135-154, (1924) · JFM 50.0371.03 [40] Kolmogoroff, A.N., Zur begründung der projektiven geometrie, Ann. math., 33, 175-176, (1932) · Zbl 0003.07803 [41] Koszul, J.L., Homologie et cohomologie des algèbres de Lie, Bull. soc. math. France, 78, 65-127, (1950) · Zbl 0039.02901 [42] Kowalsky, H.J., Zur topologischen kennzeichnung von Körpern, Math. nachr., 9, 261-268, (1953) · Zbl 0053.35401 [43] Kuratowski, C., Topologie II, (1961), Państwowe Wydawnistwo Naukowe Berlin [44] Lenz, H., Kleiner desarguesscher satz und dualität in projektiven ebenen, Jber. Deutsch. math.-verein., 57, 20-31, (1954) · Zbl 0055.13801 [45] Montgomery, D., Simply connected homogeneous spaces, (), 467-469 · Zbl 0041.36309 [46] Montgomery, D.; Zippin, L., Topological transformation groups, (1955), Wiley (Interscience) Warsaw · JFM 66.0959.03 [47] Moore, R.L., Concerning triods in the plane and the junction points of plane continua, (), 85-88 · JFM 54.0630.03 [48] Mostert, P.S., Sections in principal fiber spaces, Duke math. J., 23, 57-71, (1956) · Zbl 0072.18102 [49] Pickert, G., Projektive ebenen, (1955), Springer New York · Zbl 0066.38707 [50] Pickert, G., Eine nichtdesarguessche ebene mit einem Körper als koordinatenbereich, Publ. math. debrecen, 4, 157-160, (1956) · Zbl 0070.15604 [51] Poncet, J., Groupes de Lie compact de transformations de l’espace euclidien et LES spheres comme espaces homogènes, Comment. math. helv., 33, 109-120, (1959) · Zbl 0084.19006 [52] Pontrjagin, L., Über stetige algebraische Körper, Ann. math., 33, 163-174, (1932) · Zbl 0003.07802 [53] Pontrjagin, L.S., Topologische gruppen, () · Zbl 0085.01704 [54] Salzmann, H., Topologische projektive ebenen, Math. Z., 67, 436-466, (1957) · Zbl 0078.34103 [55] Salzmann, H., Kompakte zweidimensionale projektive ebenen, Arch. math., 9, 447-454, (1958) · Zbl 0082.35802 [56] Salzmann, H., Homomorphismen topologischer projektiver ebenen, Arch. math., 10, 51-55, (1959) · Zbl 0084.37102 [57] Salzmann, H., Topologische struktur zweidimensionaler projektiver ebenen, Math. Z., 71, 408-413, (1959) · Zbl 0092.38402 [58] Salzmann, H., Kompakte zweidimensionale projektive ebenen, Math. ann., 145, 401-428, (1962) · Zbl 0103.13503 [59] Salzmann, H., Kompakte ebenen mit einfacher kollineationsgruppe, Arch. math., 13, 98-109, (1962) · Zbl 0105.13201 [60] Salzmann, H., Zur klassifikation topologischer ebenen, Math. ann., 150, 226-241, (1963) · Zbl 0135.39201 [61] Salzmann, H., Zur klassifikation topologischer ebenen. II, Abh. math. sem. univ. Hamburg, 27, 145-166, (1964) · Zbl 0135.39201 [62] Salzmann, H., Characterization of the three classical plane geometries, Illinois J. math., 7, 543-547, (1963) · Zbl 0188.24501 [63] Salzmann, H., Zur klassifikation topologischer ebenen. III, Abh. math. sem. univ. Hamburg, 28, 250-261, (1965) · Zbl 0167.49001 [64] Salzmann, H., Polaritäten von moulton-ebenen, Abh. math. sem. Hamburg, 29, 212-216, (1966) · Zbl 0139.37804 [65] Salzmann, H., Kollineationsgruppen ebener geometrien, Math. Z., 99, 1-15, (1967) · Zbl 0146.41604 [66] Skornjakov, L.A., Topological projective planes, Trudy moskov. mat. obšč., 3, 347-373, (1954) · Zbl 0057.36201 [67] Skornjakov, L.A., Curve systems in the plane, Dokl. akad. nauk SSSR, 98, 25-26, (1954) [68] Skornjakov, L.A., Curve systems in the plane, Trudy moskov. mat. obšč., 6, 135-164, (1957) [69] Skornjakov, L.A., Metrization of the projective plane in connection with a given system of curves, Izv. akad. nauk SSSR ser. mat., 19, 471-482, (1955) · Zbl 0067.40101 [70] Stone, A.H., Sequences of coverings, Pacific J. math., 10, 689-691, (1960) · Zbl 0114.14004 [71] Tits, J., Sur LES groupes doublement transitifs continus, Comment. math. helv., 26, 203-224, (1952) · Zbl 0047.26002 [72] Tits, J., Sur LES groupes doublement transitifs continus: correction et compléments, Comment. math. helv., 30, 234-240, (1956) · Zbl 0070.02505 [73] Tschetweruchin, N.F., Eine bemerkung zu den nicht-desarguesschen linien-systemen, Jber. Deutsch. math.-verein., 36, 134-136, (1927) · JFM 53.0540.03 [74] Wagner, A., On finite affine line transitive planes, Math Z., 87, 1-11, (1965) · Zbl 0131.36804 [75] Weil, A., L’intégration dans LES groupes topologiques, (1951), Hermann Leipzig [76] Wyler, O., Order and topology in projective planes, Amer J. math., 74, 656-666, (1952) · Zbl 0047.13811 [77] Whyburn, G.T., Topological analysis, (1958), Princeton University Press Paris [78] Yaglom, I.M.; Boltyanskiǐ, V.G., Convex figures, (1961), Holt, Rinehart and Winston Princeton, New Jersey · Zbl 0098.35501 [79] Young, G.S., A generalization of Moore’s theorem on simple triods, Bull. amer. math. soc., 50, 714, (1944) · Zbl 0060.40207
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