Actions of compact groups on spheres and on generalized quadrangles

Alle Wirkungen kompakter zusammenhängender Gruppen von genügend großer Dimension auf Sphären und auf zwei Arten von verallgemeinerten Vierecken werden im einzelnen beschrieben. Für Sphären läßt sich das Ergebnis wie folgt zusammenfassen: Jede treue stetige Wirkung einer kompakten zusammenhängenden Gruppe, deren Dimension 1 + dim SO(n-2) übersteigt, auf einer n-Sphäre ist linear, also äquivalent zur natürlichen Wirkung einer Untergruppe von SO(n+1). Unter ähnlichen 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 genügend großer Dimension äquivalent 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 verallgemeinerter Polygone durch Salzmann und seine Schule. Der dabei entstandene Leitgedanke, nur die Wirkung einer Gruppe von genügend großer Dimension vorauszusetzen, wird in dieser Arbeit erstmals für verallgemeinerte Vierecke durchgeführt. Wir setzen zusätzlich voraus, daß die Gruppe kompakt ist, um die hochentwickelte Theorie der Wirkungen kompakter Gruppen auf (Kohomologie-) Mannigfaltigkeiten für die topologische Inzidenzgeometrie weiter zu erschließen. Umgekehrt ermöglicht erst die spezifische Salzmannsche Fragestellung die Ergebnisse über Sphären, die ja dem Bereich der klassischen Theorie angehören. Indem die Klassifikation der kompakten Lie-Gruppen konsequent ausgenutzt wird, läßt sich das Problem auf die Behandlung weniger Serien von Gruppen zurückführen. Bei verallgemeinerten Vierecken zeigt man dagegen zuerst die Transitivität der Wirkung und benutzt dann die bestehende (teilweise hier neu bewiesene) Klassifikation.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 dimension exceeds 1 + dim SO(n-2) on an n-sphere is linear, i.e. it is equivalent to the natural action of a subgroup of SO(n+1). Under similar hypotheses, we study actions on finite-dimensional compact generalized quadrangles whose point rows have dimension either 1 or 4. We find that every effective action of a sufficiently high-dimensional compact 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 projective 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 generalized quadrangles for the first time. Compactness of the group is a strong additional assumption which allows us to introduce the sophisticated 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 generalized quadrangles, we first show transitivity of the action and then use, and partly re-prove, recent classification results

Reports of planetary geology program, 1977-1978

A compilation of abstracts of reports which summarizes work conducted by Planetary Geology Principal Investigators and their associates is presented. Full reports of these abstracts were presented to the annual meeting of Planetary Geology Principal Investigators and their associates at the Universtiy of Arizona, Tucson, Arizona, May 31, June 1 and 2, 1978

Reports of planetary geology program, 1979 - 1980

Abstracts of 145 reports are compiled addressing the morphology, geochemistry, and stratigraphy of planetary surfaces with some specific examinations of volcanic, aeolian, fluvial, and periglacial processes and landforms. In addition, reports on cartography and remote sensing of planet surfaces are included

Finite generalized quadrangles as the union of few large subquadrangles

AbstractWe study the question: what is the smallest number n of subquadrangles of order (s,t′) of a finite generalized quadrangle Γ of order (s,t) such that the union of the point sets of all these subquadrangles is equal to the point set of Γ? It turns out that n⩾s+1 and if n=s+1, then except for a finite list of small examples, either all the subquadrangles are disjoint, or t=s=t′ and all the subquadrangles meet pairwise in a common subquadrangle of order (s,1). Examples exist in both cases and they show that a further classification is out of reach. A similar result holds for finite polar spaces

Reports of planetary geology program, 1980

This is a compilation of abstracts of reports which summarize work conducted in the Planetary Geology Program. Each report reflects significant accomplishments within the area of the author's funded grant or contract

Reports of planetary geology and geophysics program, 1989

Abstracts of reports from Principal Investigators of NASA's Planetary Geology and Geophysics Program are compiled. The research conducted under this program during 1989 is summarized. Each report includes significant accomplishments in the area of the author's funded grant or contract