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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)
##### Keywords:
intersection form; linking form; 4-manifold; presentation
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