Invent. math. 122, 531-557 (1995) Inventiones mathematicae 9 Springer-Verlag 1995 4-Manifold topology II: Dwyer's filtration and surgery kernels Michael H. Freedman j, Peter Teichner 2 i University of California, Department of Mathematics, San Diego, La .lolla, CA 92093-0112, USA; e-mail: [email protected] 2 Universitfit Mainz, Fachbereich Mathematik, D-55099 Mainz, Germany; e-mail: [email protected] Oblatum 20-11-1995 & 26-V-1995 Abstract. Even when the fundamental group is intractable (i.e. not "good") many interesting 4-dimensional surgery problems have topological solutions. We unify and extend the known examples and show how they compare to the (presumed) counterexamples by reference to Dwyer's filtration on second homology. The development brings together many basic results on the nilpotent theory of links. As a special case, a class of links only slightly smaller than "homotopically trivial links" is shown to have (free) slices on their Whitehead doubles. Introduction In dimension four, the basic machinery of manifold theory, surgery and (5-dimensional) s-cobordism theorems, exist in the topological category when the fundamental group rr is "good" [FT] and is expected to fail for 7r free (and nonabelian) and in fact to fail for the "random" group. Nevertheless, even when rc is arbitrary many special surgery problems can profitably be solved. The theorem [F2] that the Whitehead double of any boundary link is (freely) slice is an example. These applications all involve some representation of the surgery kernel by a submanifold M whose inclusion M C N into the source of the surgery problem is 7h-null. Whereas all previous applications (IF2, F3, FQ] Chapter 6) required the second homology of M to be spherical, we find here (see Theorem 1.1 and Corollary 1.2) that the important condition is only that H2(M) = ~bo,(M), i.e. that the second homology lies in the co-term of the Dwyer [D] filtration as discussed in Sect. 2. This is an important philosophical point since for any n > 1, the "canonical" (or "atomic" compare [CF]) surgery The first author is supported by the IHES, the Guggenheim foundation and the NSF. The second author is supported by the IHES and the Humboldt foundation.532 M.H. Freedman, P. Yeichner problems-to which all others restrict-can be chosen so that the kernel is car- ried by a ~l-null submanifold M with H2(M) = ~b,,(M). As elsewhere in this subject, taking the limit is the essential problem. On the way to the main theorem we develop in Sect. 2 the nilpotent theory of links and their (immersed) slices in a compact contractible 4-manifold, using only group theoretic methods (and not Massey products). This unified perspective contains many previous results (e.g. from [MI,M2, D,T,K,L or C]) but uses only the largest possible indeterminancy for the invariants. A special case of our method shows in Sect. 3 that a class of links, larger than "boundary-links" and slightly smaller than "homotopically trivial links" have (free) slices bounding their Whitehead doubles (Theorem 3.1 ). This gen- eralizes the main results of both [F2] and [F3] . 1. New surgery theorems We describe a naive (map-less) surgery theorem and then its corollary in the formal setting of normal maps. Let N be a compact connected topological 4-maniibld, possibly with bound- ary. Let M C N be a connected codimension 0 submanifold with connected boundary. Assume that M is hi-null in N, i.e. the inclusion induces the zero map ~zj(M) -+ hi(N). Assume that HI(?M) ~ HI(M). Then elementary cal- culations (see Sect. 3) show that H2(M) is tree. This says roughly that homo- logically M resembles a thickening of a 2-complex. Note that the triviality of ~I(M) ---+ zrl(N) implies a natural factorisatiou H2(M) --, 7r2(N) ~ H2(N) which we may use to define N + := NU/ff3-cells ) where the attachment is to the image in ~2(N) of a free basis [4 for H2(M). If [~1 and [~2 are two free bases they differ by a nonsingular integral linear transformation. Since any such transformation is a product of elementary matrices there exists a "deformation" of the 3-cells realizing a (simple) homotopy equivalence N/~ ~- NI~. Thus N + is well defined. It has the same 2-skeleton as N and satisfies H2(N+; 2~[zrl ]) ~ H2(M) Q-~ Z[~zl ] ~ Hz(N; 2~[ni]). The nonsingularity of the intersection form on M (see Sect. 3) makes N + a Poincar6 space, but since Theorem 1.1 puts a manifold structure directly on N +, we will not offer a separate proof for this fact. With this notation, we state a naive surgery theorem for producing a manifold with the simple homotopy type of N +. Sect. 2 treats Dwyer's [D] filtration of/42, n2 C ~,,, C ... ~h C ~b~,_l--- C q52 =//2, appearing in the statement below. Theorem 1.1. If the second homology of M C N is in the co-term oJ Dwyer's .filtration, ~,,,(M) = H2(M ) then there exists a 4-man(fbM N', with 8N' = (3N = ~N + and a (simple) homotopy equivalence (rely?) (N',t?N')~ (N+,~N+), i.e. a manifold structure (rel~) on N +.4-Manifold topology 11: Dwyer's filtration and surgery kernels 533 Now consider the formal setting of surgery. Suppose that N/~X is a de- gree 1 normal map from a topological manifold to a Poincar6 space X. There is the classical surgery obstruction 0 C L]')(nIX) to constructing a normal bor- dism to a (simple) homotopy equivalence N't~X. (We suppose here that if ~N+~/~ then .[[,w : ?N ~ ?X is already an equivalence and then the nor- mal bordism mentioned above is required to be relative to the boundary.) It is always possible to normally bord f to a =l-isomorphism with K := ker(H2(N; 2~[=IX])~ H2(X; g[nlX])) a free ~[~zlX]-module so we assume that tiffs has been done. By definition, the surgery obstruction 0 vanishes if there is a (preferred) basis for