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