LOW COHOMOGENEITY ACTIONS 5
work in Chapter II, which is necessary for the proof of Theorem A. The latter theorem
will be important since it plays, at the technical level, the role of having a "virtual"
fixed point.
In Chapter II we have, indeed, a broader perspective. First of all it is natural to
unify the G-weight pattern analysis of all actions with cohomogeneity 1 or 2. But for
simple groups this te also the appropriate place to identify those weight patterns
(irrespective of cohomogeneity) leading to principal isotropy groups of positive
dimension; this information is needed in [S7]. In fact, the final determination of G-
spheres with nontrivial principal isotropy type is still missing in the literature. Even
more, many of the technical results in Chapter II are not limited to the study of low
cohomogeneity actions; also higher cohomogeneities such as 3 and 4, should be within
reach along these lines. Again, explicit knowledge of the linear models will be crucial
since one still expects an exotic (i.e. nonlinear) but geometrically realizable weight
pattern to "resemble" a linear one.
Due to the rich variety of G-weight patterns of linear type and the existence of
exotic realizable ones as well (cf. Theorem A), it is the author's opinion that a case by
case analysis as in Chapter II can hardly be circumvented, but perhaps simplified.
Many details have been included here so that the completeness of the classification
results can be more easily checked. Indeed, Chapter II constitutes an independent and
major part which in itself justifies the present work, but we also feel it is natural to
include here (rather than in [S5]) some immediate applications, namely the results in
the last two chapters.
Chapter HI is quite short; here our purpose is mainly to give a proof of Theorem B.
Fixed point theorems of this type can be regarded as a natural generalization of the
classical P.A. Smith theory. In the latter theory explicit knowledge of the action can be
sacrificed if the (abstract) group is simple enough, whereas in the present context
similar fixed point theorems can be proved if the weight pattern of a general compact
connected Lie transformation group is reasonably simple.
In Chapter IV we shall have a closer look at cohomogeneity one G-manifolds.
Knowing the possibilities of geometric weight systems of G-spheres, it is now a rather
straightforward task to determine the possible orbit structures and prove Theorem C.
A corresponding theorem for G-spheres with dim X/G = 2 will appear in [S5]; its
proof relies heavily on Theorem A and the fact that the orbit space X/G is topologically
very simple, namely a 2-disk if G * SO(2).
Finally, we have some remarks concerning the next range of cohomogeneity, that is,
when dim X/G = 3. As was noted above, a classification of the possible weight systems
Q(X) seems to be within reach, but a description of the corresponding orbit spaces X/G
may cause trouble, due to the hard and unsolved problems in 3-dimensional topology.
Montgomery and Yang clearly faced this problem in their paper [MY;I] of 1960. But
they obtained some preliminary results in this case, under the assumption that X is the
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