Gaps in the Dimensions of Compact Transformation Groups

Abstract

Let (X, G) denote a G-space where G is a compact connected Lie group, X a connected n-dimensional manifold and the action of G on X is effective. A well-known result of Montgomery and Zippin [8] states that $\dim \;G \leqslant \;\frac{{r(r + 1)}}{2}\,\; \leqslant \;\frac{{n(n + 1)}}{2}$ where r is the maximal dimension of the orbits of G on X. In the extreme case where dimG = n(n + l)/2 it is known that G is locally isomorphic to SO(n + 1) and X is homeomorphic to either the n-sphere S ⁿ or real projective n-space P ⁿ(R) [1], [3, p. 239]. Below this maximum case there is a gap of n - 2 dimensions, at least for n ≠ 4 and n ≥ 1. In fact, we have the following result [11, p. 63], [10].

Supported in part by NSF Grant GP-5972.