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Dwyer’s filtration and topology of 4-manifolds. (English) Zbl 1047.57013

The lower central series of the fundamental group of a topological space \(X\) is closely related to the Dwyer filtration \(\phi_k(X)\) of the second homology \(H_2(X; \mathbb Z)\). It is well-known that for any \(k > 1\) the canonical \(4\)-dimensional surgery problems may be arranged to have the kernel represented by a submanifold \(M\) which is \(\pi_1\)-null, and satisfies \(H_2(M) = \phi_k(M)\) [see M. H. Freedman and F. Quinn, Topology of \(4\)-Manifolds, Princeton Mathematical Series, 39. Princeton, NJ: Princeton University Press (1990; Zbl 0705.57001)]. M. H. Freedman and P. Teichner showed in [Invent. Math. 122, 531–557 (1995; Zbl 0857.57018)] that surgery problems with an arbitrary fundamental group have a solution, provided the surgery kernel is \(\pi_1\)-null and the second homology lies in the \(\omega\)-term of the Dwyer filtration, i.e., \(H_2(M) = \phi_{\omega}(M)\). This theorem was a development of the earlier result (see Chapter 6 of the Freedman-Quinn Book) that a surgery problem can be solved if the kernel is \(\pi_1\)-null and \(H_2(M)\) is {spherical}. To state more precisely the theorem of Freedman and Teichner, it is necessary to recall the setting for surgery. Let \(N\) be a compact topological \(4\)-manifold (with or without boundary). Suppose \(f : N \to X\) is a degree \(1\) normal map from \(N\) to a Poincaré complex \(X\). Following the higher-dimensional arguments, it is possible to find a map normally bordant to \(f\) which is a \(\pi_1\)-isomorphism, and such that the kernel \[ K = \text{ker}(H_2(N; \mathbb Z[\pi_1(X)]) \to H_2(X; \mathbb Z[\pi_1(X)])) \] is a free \(\mathbb Z[\pi_1(X)]\)-module. Suppose that the Wall obstruction vanishes, so there is a preferred basis for the kernel \(K\) in which the intersection form is hyperbolic. Then we say that a compact codimension \(0\) submanifold \(M\subset \text{int} (N)\) {represents the surgery kernel} if \(M\) is \(\pi_1\)-null (i.e., the inclusion induces the trivial map on \(\pi_1\)), \(H_2(M)\) is free, and \[ H_2(M) \otimes_{\mathbb Z} \mathbb Z[\pi_1(X)] \to H_2(N; \mathbb Z[\pi_1(X)]) \] maps isomorphically onto \(K\). The Freedman-Teichner theorem says that if a standard surgery kernel is represented by \(M \subset \text{int} (N)\) which is \(\pi_1\)-null and satisfies \(\phi_{\omega}(M) = H_2(M)\), then there is a normal bordism from \(f : N\to X\) to a simple homotopy equivalence \(f' : N' \to X\). In this nice paper, the author gives a new geometric proof of the theorem of Freedman and Teichner, and analyzes its relation to the canonical surgery problems. In particular, he shows that a surgery problem with the kernel \(M\) in their setup, i.e., \(\pi_1\)-null and with \(H_2(M) = \phi_{\omega}(M)\) in a \(4\)-manifold \(N\), can be reduced to the \(\pi_1\)-null {spherical} case in the same manifold \(N\). The proof is based on the idea of splitting of capped gropes. This technique has been useful in solving a number of other problems in \(4\)-manifold topology (see the references of the paper).

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R65 Surgery and handlebodies
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