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
Previous Page Next Page