On homomorphisms between generalized polygons

Abstract

Abstract. We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphismis either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Mathematics Subject Classifications (1991): 51A25, 51 E24, 51H20, 51 H25.The aim of this paper is to investigate homomorphisms of abstract, topological, and smooth generalized polygons. The first two sections deal with generalized polygons and their homomorphisms. We have tried to make this paper self-contained, so we give a short exposition of the coordinatization and the algebraic operations (+, -, o,/) of a generalized polygon. The most important theorems at this stage are the characterization (2.7) of injective homomorphisms, and the proof (2.9) of Pasini's theorem [27] that the fibers of a non-injective homomorphism are infinite.Topological polygons are introduced in Section 3. Here, the main result is that a homomorphism is either injective or locally constant (3.4); in particular, a connected polygon admits only injective homomorphisms. For topological projective planes, this has been proved by Breitsprecher [4]. This result may be compared to Pasini's theorem [27] that finite polygons admit only injective homomorphisms.In the last section, we introduce smooth polygons. The main result of this paper states that a continuous homomorphism between smooth polygons is always a smooth embedding (4.7). In particular, every continuous automorphism of a smooth polygon is smooth, and thus the topological automorphism group of the polygon, endowed with the compact-open topology, is a smooth Lie transformation group (4.9). As an application, we prove a strong inhomogeneity theorem for a class of isoparametric hypersurfaces.

Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces

“…In view of the results in the present book, the following conjecture for the C 2 -case is very natural. (3, 4k) or (7,8).…”

“…This is essentially the method that we are going to use in the case that G is an almost simple group. For example, if G = SU (5) and H = SU(3), then we obtain m G : ( 3,5,7,9) …”

“…8.27. Multiplicities (7,8) If G acts transitively on P, then G = Spin (9) or G = Spin(9) · SO(2), and G does not act transitively on L or F . Here, P = V 2 (O 2 ) = M + .…”

“…Then either the hypersurface itself is homogeneous (and explicitly known), or it is of Clifford type with multiplicities (8,7) or (3, 4k − 4). This theorem gives in particular a new, independent proof for the classification by Hsiang and Lawson.…”

“…We apply our result about homogeneous spaces to this problem and obtain a complete classification. (8,7) or (3, 4k − 4).…”