Abstract

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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