Reducible smooth structures on 4-manifolds with zero signature
stable cohomotopy refinement of the Seiberg-Witten invariant
In this paper, the authors deal with simply-connected, zero-signature 4-manifolds. Such 4-manifolds are homeomorphic to \(n(S^2\times S^2)\) or \(m({\mathbb C}P^2\#\overline{{\mathbb C}P^2})\). In [\textit{A. Akhmedov} and \textit{B. D. Park}, J. Gökova Geom. Topol. GGT 2, 1--13 (2008; Zbl 1184.57017)] and [Math. Res. Lett. 17, No. 3, 483--492 (2010; Zbl 1275.57039)], it was proven that if an odd integer \(n\), and \(m\) is sufficiently large, then the 4-manifolds above admit infinitely many irreducible exotic smooth structures. When \(n\) and \(m\) are even integers, whether or not the 4-manifolds admit any irreducible smooth structure is an open question.NEWLINENEWLINEThis article proves that \(n(S^2\times S^2)\) and \(m({\mathbb C}P^2\#\overline{{\mathbb C}P^2})\) admit reducible smooth structures when \(n,m\) obey one of following conditions: {\parindent=6mm \begin{itemize}\item{} \(n=2p\), \(p\geq 275\), and \(p\equiv 1(2)\). \item{} \(n=2p\), \(p\geq 550\), and \(p\equiv 2(4)\). \item{} \(m=2q\), \(q\geq 51\), and \(p\equiv 1(2)\). \item{} \(m=2q\), \(q\geq 102\), and \(q\equiv 2(4)\). \item{} \(n\geq 825\) and \(n\equiv 1(4)\) \item{} \(m\geq 153\) and \(m\equiv 1(4)\) NEWLINENEWLINE\end{itemize}} Furthermore, the last two classes admit vanishing Seiberg-Witten invariants. This implies that these classes are not obtained as blow-up of certain 4-manifolds with non-vanishing Seiberg-Witten invariant. The first four classes give infinitely many non-diffeomorphic reducible smooth structures not having any metrics with positive scalar curvature. These results are also interesting when compared with the famous and classical theorem by \textit{C. T. C. Wall} [J. Lond. Math. Soc. 39, 141--149 (1964; Zbl 0131.20701)].NEWLINENEWLINETo distinguish these smooth structures, the authors use the stable cohomotopy refinement of the Seiberg-Witten invariant (Bauer-Furuta invariant) and the non-vanishing theorem. As for the construction of manifolds, they use Akhmedov, Park, and Hughes's irreducible symplectic smooth structure of \((2q-1)({\mathbb C}P^2\#\overline{{\mathbb C}P^2})\), \((2p-1)(S^2\times S^2)\), and Fintushel-Stern's knot surgery. The reference papers are the two papers in the first paragraph and [\textit{A. Akhmedov} et al., Pac. J. Math. 261, No. 2, 257--282 (2013; Zbl 1270.57067)]. These manifolds contain a square zero symplectic 2-torus, and fibered knot surgeries along the torus give symplectic manifolds again. This construction causes the existence of infinite smooth structures.NEWLINENEWLINEThe authors also discuss reducible exotic smooth structures that are not homeomorphic to manifolds with non-zero signature.
scientific article; zbMATH DE number 6349714