CLASSIFICATION OF CLOSED TOPOLOGICAL 4-MANIFOLDS

The big breakthrough in the classification of topological 4-manifolds certainly was Freedman’s proof of the topological dics embedding theorem for poly-finite or-cyclic fundamental groups in 1982 [1]. It implies the topological s-cobordism and surgery theorems for the same class of fundamental groups. Freedman deduced that the intersection form on H2 and the Kirby-Siebenmann invariant classify simply connected closed 4-manifolds up to homeomorphism and that every unimodular symmetric bilinear form is realized as the intersection form of such a manifold. In this note we are going to describe some results in the nonsimply connected case. We have to stick to the above class of fundamental groups because the main problem of whether the topological disc embedding theorem holds in general is still unsolved. (It is nowadays known as the A-B-slice problem.) In 1988, Hambleton and Kreck [3] proved a word by word generalization of Freedman’s classification result to the case of a cyclic fundamental group of odd order. They had to introduce several new techniques into the field because a priori one has to work with the equivariant intersection form on π2 to even obtain a homotopy classification. This difficulty can be understood very well in the case of a