Henceforth, to be more precise we shall assume that G is a compact connected Lie
group and G-manifolds are differentiable (C°°). In 1956 Montgomery-Samelson-Yang
[MSY] proved that if M is euclidean n-space and dim M/G = 1 or 2, then the action is
equivalent to a linear action. As a consequence, they obtained a proof of the
corresponding regularity theorem for G-spheres S
in the special case that G has a fixed
point, namely the action is equivalent to an orthogonal one in this special case.
We now turn to the final classification of compact connected transformation groups
of spheres with cohomogeneity one. Also in the abscence of fixed points, some
preliminary results were already obtained by Mostert [M] and Nagano [Ng], but a
systematic classification was first studied in a paper of 1960 by H. C. Wang [W]. He
proved the orthogonality of such transformation groups on S n under the dimension
restriction that n is even but not equal to 4, or n is odd and greater than 31. However,
as it was pointed out in [HH ], the above classification result had, unfortunately,
overlooked a rather interesting family of non-orthogonal actions and thus needed both a
correction of its result and a careful reexamination and tightening of its proof.
Following the ideas of Wang, in 1981 T. Asoh extended Wang's results to (Z
homology) spheres of dimension n * 7, and later gave a proof for the intricate case
n = 7, see [A]. One of the purposes of the present paper is to accomplish a complete
classification of cohomogeneity one transformation groups on spheres of all dimensions
via an approach entirely different from that of [W] and [A].
Next, let us focus attention on the classification of cohomogeneity 2 transformation
groups on spheres. In two papers of 1965, G. Bredon [Br1,Br2] obtained a
classification under the restrictive condition of only two types of orbits. The final
result of this rather special case, in fact, already includes a family of interesting non-
orthogonal examples. More recently, there are some further studies of the classification
problem, with narrowly specified groups and orbit structures, by Uchida-Watabe [W]
and Nakanishi [Nk]. In this paper and its succeeding sequel II (see [S5]), we shall give a
classification of all compact connected differentiable transformation groups on
homotopy spheres with cohomogeneity 2.
The major step in our classification of transformation groups on spheres with low
cohomogeneity is the complete determination of all the possibilities of the groups
together with their orbit structures. Although there are already known examples of
non-orthogonal transformation groups on spheres of cohomogeneities both 1 and 2, it is
still quite reasonable to expect that the orbit structure of any such transformation
group should closely resemble that of a suitable orthogonal one. Therefore, the first
preparatory step is, of course, the classification of all orthogonal actions on spheres of
cohomogeneities 1 or 2 and the computation of their orbit structures.The final results
of such a classification of linear groups are listed in the Tables at the end of Chapter I.
Previous Page Next Page