A new technique for the link slice problem

Abstract

The conjectures that the 4-dimensional surgery theorem and 5-dimensional s-cobordism theorem hold without fundamental group restriction in the topological category are equivalent to assertions that certain "atomic" links are slice. This has been reported in [CF, F2, F4 and FQ]. The slices must be topologically flat and obey some side conditions. For surgery the condition is: ~a(S 3 ~ slice)--, rq (B 4 slice) must be an epimorphism, i.e., the slice should be "homotopically ribbon"; for the s-cobordism theorem the slice restricted to a certain trivial sublink must be standard. There is some choice about what the atomic links are; the current favorites are built from the simple "Hopf link" by a great deal of Bing doubling and just a little Whitehead doubling. A link typical of those atomic for surgery is illustrated in Fig. 1. (Links atomic for both s-cobordism and surgery are slightly less symmetrical.) There has been considerable interplay between the link theory and the equivalent abstract questions. The link theory has been of two sorts: algebraic invariants of finite links and the limiting geometry of infinitely iterated links. Our object here is to solve a class of free-group surgery problems, specifically, to construct certain slices for the class of links ~ where D(L)eCg if and only if D(L) is an untwisted Whitehead double of a boundary link L. Unfortunately, (~ appears not to be atomic for surgery or s-cobordism. However our technique is quite different from the earlier extension [F4] of simply-connected methods and may prove helpful in establishing the limits of the nonsimply-connected theory. An n-component link is a topological imbedding I ) Si c L , $3 of n circles i = 1 in the 3-sphere. It is slice if and only if there is a topological imbedding L (dotted arrow) making the diagram: