The cohomogeneity of a transformation group (G,X) is, by definition, the dimension
of its orbit space, c = dim X/G. By enlarging this simple numerical invariant, but
suitably restricted, one gradually increases the complexity of orbit structures of
transformation groups. This is a natural program for classical space forms, say, which
traditionally constitute the first canonical family of testing spaces, due to their unique
combination of topological simplicity and abundance in varieties of compact
differentiable transformation groups.
The restrictive subfamily of linear (or linearly induced) actions on classical spaces
already exhibit a great amount of variation of orbit structures. Therefore, one expects
that more general differentiable compact connected Lie transformation groups, of low
cohomogeneity at least, should resemble those natural ones in many basic features. For
example, for euclidean G-spaces with c 2 it has been known since 1956 that the
action must be equivalent to a linear one. In this paper we address the corresponding
problem for G-spaces which are homotopy spheres.
The lack of fixed point is circumvented by the notion of geometric weight system,
which is based upon the generalized Schur's splitting principle at characteristic class
level in equivariant cohomology. The geometric weight system is determined for all G-
spheres with c 2. The case c = 1 is also supplemented with a new and complete proof
of the classification of all G-spheres up to equivalence.
Knowledge of all possible weight systems is also the crucial information we need to
classify all G-spheres when c = 2. This work is continued in the succeeding part II
which will appear in a separate issue of Memoirs in late 1996.
Key words and phrases. Lie transformation groups, spherical G-manifolds, low
cohomogeneity, linear and nonlinear actions, geometric weight system.
Received by the editor January 29, 1993; and in revised form December 22, 1993.
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