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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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