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Homology lens spaces in topological \(4\)-manifolds. (English) Zbl 1086.57018

The main result is that every homology lens space embeds topologically locally flatly in each of \(\#_8\mathbb CP^2\), \(\#_2(\mathbb CP^2\#\overline{\mathbb CP^2})\) and \(\#_4(S^2\times{S^2})\). The key algebraic step shows that the torsion linking pairing on \(H_1(L(p,q);\mathbb Z)\) is the boundary of an odd symmetric pairing on \(\mathbb Z^2\) and of an even symmetric pairing on \(\mathbb{Z}^4\). The arguments are direct and short, although ultimately depending on the fact that homology 3-spheres bound contractible 4-manifolds and on the classification of 1-connected 4-manifolds up to homeomorphism.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N35 Embeddings and immersions in topological manifolds