4-Manifolds with large symmetry groups

Abstract

The restriction to non-abelian Lie groups avoids the difficult problem of classifying 4manifolds with S’ or T ‘-actions [3,4,9-l 11. (It is well known that any 4-manifold with an effective T “-action, n 2 3, is T 4 or St x L, where L is a lens space.) This paper thus reduces the general classification problem for closed, orientable 4-manifolds with a compact Lie group G of symmetries to the cases G = S’ or T2. In $1 we show that one need only consider actions of SU (2) or SO (3), and the well known subgroup structure of these two groups is recalled. A complete equivariant classification of SU (2) and SO(3)-actions on 4-manifolds (including the non-orientable case) is given in 9 2. The codimension 1 case is taken largely from the second author’s thesis [ 123. The final section gives the topological classification in the orientable case. This leaves open the non-orientable case, where the situation seems quite interesting and more intricate (cf. [13] for the codimension 1 case). We plan to take this up in a sequel to this paper. We shall work in the smooth category. B” will denote the n-ball, s” the n-sphere, and P” the real projective n-space. The reader is referred to Bredon’s book [l] for the basic definitions and theorems of transformation groups.