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We consider surfaces of the form F U/c D where F is a Seifert surface and D is a slicing disk for the knot K. We show that, in general, there is no 3-manifold M which spans F Uk L> in the 4-ball such that F can be compressed to a disk in M. This note is concerned with the following problem. Suppose K is a classical knot which is spanned by the Seifert surface F in S3 = dB4 and the slice disk D in B4. Must the surface FUk D bound a smoothly embedded, compact, orientable 3-manifold M (also referred to as a Seifert manifold) in which F is compressible to a disk? This problem arises quite naturally from the proof that slice implies algebraically slice. Indeed, the proof that slice implies algebraically slice boils down to verifying that given any M as above (such an M always exists by transversality) a 1/2-basis of Hi (F; Q) can be realized by curves in F which are rationally null-homologous in M. The above problem is equivalent, by Stallings' version of the Loop Theorem [S], to asking if an M can be found in which a 1/2-basis for Hx(F) is realized by curves in F which are null-homotopic in M. We establish in §2 that the answer to this problem is in general no. In fact, by appropriately modifying Zeeman's [Z] notion of twist spinning an arc, a collection of surfaces of the form F Uk D are constructed in § 1 for which we give necessary and sufficient conditions on F L¡k D in order for it to bound an M in which F can be compressed to a disk. These conditions are geometric in nature—yet, they are applicable. 1. The examples. This section is devoted to the construction of particular surfaces of the form F U/f D. The pertinent properties of these surfaces will be postponed until the next section. We begin the construction of the desired surfaces with the following data: (1) K' is a smooth knot in S3, (2) F' is an incompressible Seifert surface for K', (3) Nl(K'), i = 1,2, denote tubular neighborhoods of K' where Ni(K') C intN2(K'), and (4) n is a positive integer. Now let K' — A U B where A and B are compact, connected subarcs of K' satisfying (int A) fl (inti?) = 0. We write N\(K') as the obvious decomposition N(A) U N(B); see Figure 1. Note that A4 = [S3 int N(B)} x / is the 4-ball. Received by the editors September 24, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25, 57Q45. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

A note concerning the 3-manifolds which span certain surfaces in the 4-ball