Compact Connected Lie Transformation Groups On Spheres With Low Cohomogeneity
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Compact Connected Lie Transformation Groups on Spheres with Low Cohomogeneity. II
Author: Eldar Straume
language: en
Publisher: American Mathematical Soc.
Release Date: 1997
The cohomogeneity of a transformation group ([italic capitals]G, X) is, by definition, the dimension of its orbit space, [italic]c = dim [italic capitals]X, G. We are concerned with the classification of differentiable compact connected Lie transformation groups on (homology) spheres, with [italic]c [less than or equal to symbol] 2, and the main results are summarized in five theorems, A, B, C, D, and E in part I. This paper is part II of the project, and addresses theorems D and E. D examines the orthogonal model from theorem A and orbit structures, while theorem E addresses the existence of "exotic" [italic capital]G-spheres.
Compact Connected Lie Transformation Groups on Spheres with Low Cohomogeneity, I
Author: Eldar Straume
language: en
Publisher: American Mathematical Soc.
Release Date: 1996
The cohomogeneity of a transformation group ([italic capitals]G, X) is, by definition, the dimension of its orbit space, [italic]c = dim [italic capitals]X, G. By enlarging this simple numerical invariant, but suitably restricted, one gradually increases the complexity of orbit structures of transformation groups. This is a natural program for classical space forms, which traditionally constitute the first canonical family of testing spaces, due to their unique combination of topological simplicity and abundance in varieties of compact differentiable transformation groups.
Compact Connected Lie Transformation Groups on Spheres with Low Cohomogeneity, I
Author: Eldar Straume
language: en
Publisher: American Mathematical Soc.
Release Date: 1996-02-05
In the study of Lie transformation groups on classical space forms, one of the most exciting features is the existence of nonlinear or ``exotic'' actions. Among the many unsolved problems, the classification of G-spheres with 2-dimensional orbit space has a prominent place. The main purpose of this monograph is to describe the beginnings of a program to the complete solution of this problem. One major feature of the author's approach is the effectiveness of the geometric weight system, which was introduced by Wu-Yi Hsiang around 1970, as a book-keeping method for orbit structural data. Features: Complete tables of compact connected linear groups of cohomogeneity $< 3$. Geometric weight systems techniques. Complete classification of G-spheres of cohomogeneity one. Weight classification of G-spheres of cohomogeneity two, the crucial step of the complete classification for cohomogeneity two.