Actions of the torus on $4$-manifolds. I

Abstract

Smooth actions of the 2-dimensional torus group SO(2) x 50(2) on smooth, closed, orientable 4-manifolds are studied. A cross-sectioning theorem for actions without finite nontrivial isotropy groups and with either fixed points or orbits with isotropy group isomorphic to SO(2) yields an equivariant classification for these cases. This classification is made numerically specific in terms of orbit invariants. A topological classification is obtained for actions on simply connected 4-manifolds. It is shown that such a manifold is an equivariant connected sum of copies of complex projective space CP2, -CP2 (reversed orientation), S2 x S2 and the other oriented S2 bundle over S2. The latter is diffeomorphic (but not always equivariantly diffeomorphic) to CP2 # -CP2. The connected sum decomposition is not unique. Topological actions on topological manifolds are shown to reduce to the smooth case. In an appendix certain results are extended to torus actions on orientable 4dimensional cohomology manifolds.