Let Mn be an n-dimensional smooth manifold in the (n + 2)-sphere Sn + 2, and let Sn + 2 – N(Mn) be the complement of an open regular neighbourhood of Mn in Sn + 2. A Seifert manifold H of Mn is a compact orientable (n + 1)-dimensional submanifold of Sn + 2 such that the boundary of H is Mn. Moreover, if the induced homomorphism from π1(H) into π1(Sn + 2 – N(Mn)) is one-to-one, then the Seifert manifold H is called minimal.

One important case is when Mn is a sphere, i.e. a knot. The existence of Seifert manifolds of knots of any dimension is known. It is well known that the Seifert surfaces of 1-knots with minimal genus are minimal. Gutierrez [1] asserted the existence of minimal Seifert manifolds for knots of any dimension. However, his proof has a gap. For n ≥ 3, Silver [6] has given examples of n-dimensional knots with no minimal Seifert manifolds. Necessary and sufficient conditions for an n-knot (n ≥ 3) to have a minimal Seifert manifold have been given in [7].

In this paper we prove the following result:

Theorem. For any integer g ≥ 1, there exist closed orientable surfaces of genus g in S4 which have no minimal Seifert manifolds and no trivial 1-handles.

This result was presented to KOOK topology seminar in Osaka in September 1989, and at KNOT 90 in Osaka in 1990. The existence of 2-knots with no minimal Seifert manifolds is still an open question.