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Construction of compact 4-manifolds with infinite cyclic fundamental groups. (English) Zbl 0911.57015
M. H. Freedman [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)] and F. S. Quinn [Topology of 4-manifolds, Princeton Math. Ser. 39 (1990; Zbl 0705.57001)] gave a classification of closed simply connected 4-manifolds in terms of their intersection forms and Kirby-Siebenmann invariants. This was generalized by S. Boyer [Trans. Am. Math. Soc. 298, 331-357 (1986; Zbl 0615.57008); Comment. Math. Helv. 68, No. 1, 20-47 (1993; Zbl 0790.57009)] to the case of simply connected manifolds with boundary. In particular, Boyer proves a realization theorem which says that whenever \(M\) is a closed connected 3-manifold with linking form \(\l_M\) on \(H_1(M;Z)\) and \(\mathcal L\) is a symmetric bilinear pairing on a free abelian group \(E\) for which there is a presentation of \(H_*(M,Z)\) by \((E,\mathcal L),\) then there is a simply connected 4-manifold \(V\) bounded by \(M\) with the intersection form isomorphic to \((E,\mathcal L).\) The concept of a presentation algebraicizes the structure coming from the long exact sequence of the pair \((V,M)\) when it exists, including the relation of the linking form on \(M\) and the intersection form on \(V.\) Boyer classifies simply connected 4-manifolds with boundary \(M\) in terms of the set of all presentations, Kirby-Siebenmann invariants, and spin structures on the boundary. In this paper the author utilizes Boyer’s techniques to construct a 4-manifold \(V\) with infinite cyclic fundamental group whose boundary \(M\) is a given homology \(S^1 \times S^2\) and whose intersection form is isomorphic to a given Hermitian form over \(\Lambda = Z[Z]\) so that the inclusion \(\pi_1(M) \to \pi_1(V)\) is surjective. The definitions involve the appropriate generalization of the notion of a presentation to this context, which is what is required algebraically when one has such a 4-manifold and embodies the relationship of the intersection form of \((V,M)\) over \(\Lambda\) with the linking pairing on \(M.\) The construction of \(V\) starts from the presentation and first forms a cobordism using a framed link construction from \(M\) to \(N\) and utilizes a result of Freedman and Quinn to cap off \(N\) with a homotopy circle to get \(V.\)

MSC:
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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