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Actions of Compact Groups on Spheres and on Generalized Quadrangles Actions of Compact Groups on Spheres and on Generalized Quadrangles Von der Fakult¨at Mathematik der Universit¨at Stuttgart zur Erlangung der Wurde¨ eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung Vorgelegt von Harald Biller aus Frankfurt am Main Hauptberichter: Privatdozent Dr. Markus Stroppel Mitberichter: Prof. Dr. Hermann H¨ahl Prof. Dr. Theo Grundh¨ofer (Universit¨at Wurzburg)¨ Tag der mundlichen¨ Prufung:¨ 8. Juni 1999 Mathematisches Institut B der Universit¨at Stuttgart 1999 ii Harald Biller Fachbereich Mathematik Technische Universit¨at Darmstadt Schloßgartenstraße 7 D-64289 Darmstadt Germany [email protected] Mathematics Subject Classification (1991): 51H10 Topological linear incidence structures 51E12 Generalized quadrangles, generalized polygons 57S10 Compact groups of homeomorphisms 57S25 Groups acting on specific manifolds 57P99 Generalized manifolds Keywords: topological geometry, generalized quadrangle, compact transformation group, homology sphere, cohomology manifold This thesis is available online: http://elib.uni-stuttgart.de/opus/volltexte/1999/566 Abstract The actions of sufficiently high-dimensional compact connected groups on spheres and on two types of compact Tits buildings are classified explicitly. The result for spheres may be summarized as follows: every effective continuous action of a compact connected group whose di- mension exceeds 1 + dim SOn 2R on an n-sphere is linear, i.e. it is − equivalent to the natural action of a subgroup of SOn+1R. Under sim- ilar hypotheses, we study actions on finite-dimensional compact gener- alized quadrangles whose point rows have dimension either 1 or 4. We find that every effective action of a sufficiently high-dimensional com- pact group is equivalent to an action on a Moufang quadrangle, i.e. on a coset geometry associated to a BN-pair in a simple Lie group. Both for spheres and for generalized quadrangles, the classification arises from an explicit description of the actions. One main source for this thesis is the investigation of compact pro- jective planes and, recently, other compact generalized polygons by Salzmann and his school. They developed the specific hypothesis of a sufficiently large group dimension, which here is applied to gener- alized quadrangles for the first time. Compactness of the group is a strong additional assumption which allows us to introduce the sophisti- cated theory of actions of compact groups on (cohomology) manifolds further into topological incidence geometry. Conversely, the results about spheres, which lie completely within the scope of the classical theory, are rendered possible by Salzmann's specific question. When combined with a thorough exploitation of the classification of compact Lie groups, it essentially reduces the problem to the consideration of a small number of series of groups. To obtain the results about gen- eralized quadrangles, we first show transitivity of the action and then use, and partly re-prove, recent classification results. Zusammenfassung Alle Wirkungen kompakter zusammenh¨angender Gruppen von genu-¨ gend großer Dimension auf Sph¨aren und auf zwei Arten von verallge- meinerten Vierecken werden im einzelnen beschrieben. Fur¨ Sph¨aren l¨aßt sich das Ergebnis wie folgt zusammenfassen: Jede treue steti- ge Wirkung einer kompakten zusammenh¨angenden Gruppe, deren Di- mension 1 + dim SOn 2R ubersteigt,¨ auf einer n-Sph¨are ist linear, also − ¨aquivalent zur naturlichen¨ Wirkung einer Untergruppe von SOn+1R. Unter ¨ahnlichen Voraussetzungen untersuchen wir Wirkungen auf endlichdimensionalen kompakten verallgemeinerten Vierecken, deren Punktreihen Dimension 1 oder 4 haben. Hier zeigen wir, daß jede treue Wirkung einer kompakten Gruppe von genugend¨ großer Dimen- sion ¨aquivalent ist zu einer Wirkung auf einem Moufang-Viereck, also auf einer Nebenklassengeometrie einer einfachen Lie-Gruppe, die durch ein BN-Paar beschrieben wird. Die vorliegende Arbeit steht in der Tradition der Untersuchung kompakter projektiver Ebenen und neuerdings anderer kompakter ver- allgemeinerter Polygone durch Salzmann und seine Schule. Der dabei entstandene Leitgedanke, nur die Wirkung einer Gruppe von genugend¨ großer Dimension vorauszusetzen, wird in dieser Arbeit erstmals fur¨ verallgemeinerte Vierecke durchgefuhrt.¨ Wir setzen zus¨atzlich voraus, daß die Gruppe kompakt ist, um die hochentwickelte Theorie der Wir- kungen kompakter Gruppen auf (Kohomologie-) Mannigfaltigkeiten fur¨ die topologische Inzidenzgeometrie weiter zu erschließen. Umge- kehrt erm¨oglicht erst die spezifische Salzmannsche Fragestellung die Ergebnisse uber¨ Sph¨aren, die ja dem Bereich der klassischen Theo- rie angeh¨oren. Indem die Klassifikation der kompakten Lie-Gruppen konsequent ausgenutzt wird, l¨aßt sich das Problem auf die Behand- lung weniger Serien von Gruppen zuruckf¨ uhren.¨ Bei verallgemeinerten Vierecken zeigt man dagegen zuerst die Transitivit¨at der Wirkung und benutzt dann die bestehende (teilweise hier neu bewiesene) Klassifika- tion. Contents List of Tables vii Preface ix Deutschsprachige Kurzfassung xv 1 Foundations 1 1.1 Locally compact groups .................. 2 1.2 Cohomology manifolds ................... 4 1.3 Generalized spheres ..................... 12 1.4 Compact quadrangles ................... 14 2 Actions of compact groups 23 2.1 General theory ....................... 23 2.2 Actions on cohomology manifolds ............. 36 2.3 Consequences of a theorem of Mann's .......... 38 2.4 Calculating homology ................... 41 2.5 Large subalgebras of compact Lie algebras ........ 49 3 Spheres 63 3.1 Very large orbits ...................... 63 3.2 Complex unitary groups .................. 73 3.3 A homogeneity property of torus groups ......... 83 3.4 Other particular groups .................. 85 3.5 Local type of compact Lie transformation groups .... 91 3.6 Reconstruction of actions ................. 102 v vi CONTENTS 4 Compact (1; m)-quadrangles 115 4.1 Miscellaneous tools for compact polygons ........ 115 4.2 Line-homogeneous quadrangles .............. 119 4.3 Local type of compact Lie transformation groups .... 130 4.4 Reconstruction of actions ................. 145 5 Compact (4; m)-quadrangles 151 5.1 Compact Lie groups .................... 152 5.2 Compact non-Lie groups .................. 161 5.3 Characterization theorems ................. 182 Bibliography 187 Notation 201 Index 203 List of Tables 2.1 The simple compact Lie algebras and their subalgebras of maximal dimension ................... 50 2.2 Structure of semi-simple compact Lie groups acting on spaces of small dimension ................. 60 3.1 The semi-simple Lie algebras of rank at most 5 ..... 78 3.2 The compact Lie algebras for effective actions on S8 .. 94 3.3 Actions of large compact connected groups on n-spheres 114 5.1 Large compact Lie algebras of given rank ........ 153 vii viii Preface To study an object is to study its symmetry { this thought is the heart of Felix Klein's influential Erlangen Programme. In the present thesis, it is applied to topological spheres and to two classes of topological Tits buildings. In each setting, the actions of sufficiently high-dimensional compact connected groups are determined. The results are most eas- ily formulated for spheres. The group SOn+1R acts naturally on the n-dimensional sphere Sn. If a compact connected group whose dimen- sion is greater than that of SO2R SOn 2R acts continuously and × − effectively on Sn then the action is equivalent to the natural action of a subgroup of SOn+1R. The proof combines the theory of Lie groups with algebraic topology. The Tits buildings which we study are generalized quadrangles. These geometries consist of points and lines. Their characteristic prop- erty is that for every non-incident point-line pair, there is a unique incident pair such that the four elements form a chain. We assume that this chain depends continuously on its ends with respect to some finite-dimensional compact topology, and that the points which lie on any one line form sets of dimension either 1 or 4. Each effective action of a sufficiently high-dimensional compact connected group on such a generalized quadrangle is equivalent to an action on a Moufang quad- rangle, i.e. on one which is associated to a BN-pair in a simple Lie group; the group, the quadrangle, and the action are described ex- plicitly. In particular, we obtain sharp upper bounds for the group dimensions. The interest in generalized quadrangles is the reason for working in the category of continuous actions. In the absence of differentiability assumptions, we are referred to the theory of topological transforma- tion groups which flourished in the context of the solution of Hilbert's ix x PREFACE Fifth Problem by Montgomery and Zippin. Their work, as well as contributions by Borel, Bredon, and others, is essential. The second main source of inspiration for this thesis, and in fact its original motivation, is the work of Salzmann on compact projective planes, which are the smallest non-trivial compact buildings. Salz- mann and his school developed the use of the group dimension as the principal measure of symmetry, and many of their methods are relevant in our context. More recently, the investigation of general topological buildings was begun by Burns, Spatzier, and Thorbergsson in differ- ential geometry, and by Forst, Grundh¨ofer,Joswig, Knarr, Kramer, Schroth, the Stroppels, and Van Maldeghem in Salzmann's tradition. A more detailed description of the results will now be given in con- nection with an overview of the
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