3-dimensional homology handles and minimal second Betti numbers of 4-manifolds
The present paper faces the problem of computing the minimal second Betti number of compact 4-manifolds bounded by a given closed 3-manifold \(M\). Note that if the bounding 4-manifolds are assumed to be simply connected, then the problem may be solved via the Boyer classification theorem [\textit{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)]. Here, on the contrary, the author takes into consideration 4-manifolds whose fundamental group is isomorphic to an infinite cyclic group; in particular, the analysis is performed in case of \(M\) being a homology handle (i.e. an orientable closed 3-manifold with the same integral homology groups as \(\mathbb{S}^1\times\mathbb{S}^2\): see [\textit{A. Kawauchi}, Osaka J. Math. 12, 565-581 (1975; Zbl 0315.57002)] such that the homomorphism of \(\pi_1\) induced from the inclusion \(i_\#: \pi_1(M) \to \mathbb{Z} \) is surjective. With these hypotheses, let \(\beta^{\text{TOB}}(M)\) and \(\beta^{ \text{DIFF}}(M)\) denote the minimal second Betti number of bounding 4-manifolds in the topological and smooth category respectively; the main result of the present paper states that, for any positive integer \(n\), there exist infinitely many distinct homology handles \(\{M_m^{(n)}\}_{m\in \mathbb{N}}\) -- directly obtained by 0-surgery on suitable knots -- with \(\beta^{\text{TOP}} (M_m^{(n)}) =\beta^{ \text{DIFF}}(M_m^{(n)}) =n\). Moreover, a class \({\mathcal K}\) of knots is detected, such that a gap between \(\beta^{\text{TOP}}\) and \(\beta^{\text{DIFF}}\) holds for every homology handle obtained by 0-surgery on \(K\in{\mathcal K}\).
scientific article; zbMATH DE number 1216667