T The cohomogeneity of a compact (differentiable) transformation group (G,M) is, by definition, the codimension of its principal orbits, which is clearly also equal to the dimension of its orbit space M/G. For example, transformation groups of cohomogeneity zero are exactly the transitive ones and transformation groups of cohomogeneities 1, 2, . . are, therefore, natural gradual extensions of the transitive ones. The study of manifolds with symmetry groups has always been a central and fruitful avenue of research in topology and the general study of manifolds. From this point of view, a G- manifold M with low cohomogeneity is certainly a manifold with a given large group G of symmetries. In low-dimensional topology, for example, it is well known that certain S - manifolds are basic building blocks in various structure theorems for 3-dimensional manifolds. Similar phenomena in dimension 4 or higher, involving more general groups, are also of current interest and are generally attacked from the viewpoint of global transformation groups, see e.g. Melvin-Parker [MP] in dimension 4. In fact, the general pattern seems to be that, as long as dim M/G 2 and the orbital stratification of M/G is well understood, one should be able to reconstruct what are the possible pairs (G,M). On the other hand, once G is given and we assume dim M/G 2, it is also natural to remove any dimension restriction on M. Our attribute low cohomogeneity refers to this category of G-manifolds. Compact transitive transformation groups on spheres were classified in consecutive papers of Montgomery- Samelson [MS] and A. Borel [B1], and the final result is simply that all compact transitive transformation groups on spheres are conjugate to the orthogonal ones. The general study of G-manifolds with low cohomogeneity 0 dates back to the earliest works on (topological) transformation groups say, by Montgomery, Samelson and Zippin, starting around 1940. Since those early days, the classical space forms such as euclidean spaces, spheres and projective spaces have been the most popular testing spaces, both for their topological simplicity and the fact that they accomodate a rich variety of well-known and natural transformation groups. In 1941 Montgomery and Zippin (cf.[MZ1]) first studied compact connected transformation groups on the n-sphere S n with an (n-l)-dimensional orbit. When G has a fixed point, they determined the orbits and reduced the problem to actions on euclidean n-space and to transitive actions on homotopy spheres. Although Montgomery and Zippin also proved (cf. [MZ2]) that compact connected groups on euclidean 3-space are equivalent to groups of linear transformations, the difficulties in attempting to prove similar linearization or regularity results in higher dimensions were clearly recognized at that time. 1
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