standard n-sphere, n 4 and G has a fixed point. Namely, X/G is a simply connected 3-
manifold with S
as boundary. They also observed that if X/G actually is the 3-disk,
then the action is equivalent to an orthogonal action. It seems to be an open problem
whether their assumption on the topology of X/G is necessary. However, in the abscence
of fixed points, when exotic actions definitely must exist, it is a much harder and
unsolved problem to describe all possibilities.
In retrospect, during the last 10-15 years there have been many applications of
low cohomogeneity actions in differential geometry, notably by Wu-yi Hsiang and his
collaborators, cf. e.g., [Hs2]. We also mention some recent activity in algebraic
geometry, by the school of H. P. Kraft and G. W. Schwarz, concerning algebraic actions
of reductive groups. For example, cohomogeneity one actions on affine acyclic varieties
are investigated in [KS]. It also seems that the concept of geometric weight system and
various results from our differential topological setting can be adapted to the study of
reductive group actions on (acyclic) varieties, see e.g., [F] for new results in this area.
A preprint of the present work has been circulating in the mathematical community
since the late 1980's. Its (somewhat overdue) publication has been encouraged by the
recent interest in the methodology of low cohomogeneity. The author is particularly
indebted to Wu-yi Hsiang for his kind advice and constant support through many years.
Notation and terminology
We try to be consistent and use "standard" notation wherever possible, cf. e.g., [Bo2],
[Br3], [Hs1]. We give some examples.
If H = G, then N Q ( H ) (or simply N(H)) denotes the normalizer of H in G; Z(H) (esp.
Z Q ( H ) ) is the center of H (resp. centralizer of H in G)). The fixed point set of G in X is
written as F(G), F(G,X) or X
To indicate that X is a compact differentiable manifold with the same A-cohomology
as the n-sphere, we write X~^ S
. Here A is the integers 7L or rationals (Q,but we
write X~p S
if A is the prime field Z
of order p. The quaternions are denoted by H.
A cyclic group of order k is usually written Z
, and Dk is the dihedral group of order
2k. For nonzero integers r and s their greatest common divisor is (r,s) 1.
*¥$ is the complexification of a real representation, and [E]K is the realification of
a complex or quatemionic representation O. The trivial 1-dimensional representation
(over (R) is denoted either by 1 or x\. (T^ = d-q is d-dimensional and trivial.)
The longer chapters are divided into sections §1, §2, . . Typically, Lemma 2.5
refers to §2 (of a specific chapter), but within each section more local formulas or
statements are labelled consecutively as (1), (2) etc.
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