A 2-knot is a locally flat embedding K : S2 → S4. The closed 4-manifold M(K) obtained from S4 by surgery on K is orientable, has Euler characteristic 0 and πK = π1(M(K)) has weight 1 (i.e., is the normal closure of a single element) and infinite cyclic abelianization, πK/πK′ = H1(M(K);Z) = Z. Conversely, if M is a closed orientable 4-manifold with χ(M) = 0 and π1(M) of weight 1 then it may be obtained in this way, for surgery on a loop in M representing a normal generator for π1(M) gives a 1-connected 4-manifold Σ with χ(Σ) = 2 which is thus homeomorphic to S4 and which contains an embedded 2-sphere as the cocore of the surgery. If π1(M) is solvable then it has weight 1 if and only if π1(M)/π1(M)′ is cyclic, for a solvable group with trivial abelianization must be trivial. (See [H] for more details).

In the next two sections we shall summarize progress made on the study of 2-knot groups since the appearance of [H]. We shall then consider when the manifolds M(K) admit geometries or complex analytic structures. The final section gives new characterizations of minimal complex surfaces which are ruled over curves of genus > 1 or are elliptic surfaces, fibred over such curves.