Toral Actions on 4-manifolds and Their Classifications

P. Orlik and F. Raymond showed, in [OR, I], the following: Suppose that M is a ^-dimensional closed simply-connected manifold with an effective T1-action. Then M is an equivariant connected sum of CP2, CP2, S2x S2 and S4. In [OR, II], they studied some non-simply-connected manifolds with an effective T2-action and proved that, if the manifolds have neither fixed points nor circle subgroups as stabilizers, then, in "almost air cases, two manifolds are diffeomorphic if and only if they are equivariantly diffeomorphic up to an automorphism of T2. With the presence of fixed points or circle orbits, the techniques of a topological classification are quite different. Orlik and Raymond obtained an equivariant classification when M has a fixed point, but there were three families of T2 -manifolds, called basic blocks. To obtain a topological classification of closed orientable T2-manifolds with a fixed point, it was necessary to study these families of T2-manifolds, which are described in terms of orbit spaces. For example, each of the manifolds of one family has the orbit space pictured in Figure 1 (see §2 for a description of this space and [OR, II]). They showed that M#k(S2 x S2) = (S1 x S3)#(S2 x S2)#k(S2 x S2), if mn is even,