Seiberg-Witten invariants of 4-manifolds with free circle actions (Q2764511)

Let \(X\) be a smooth 4-manifold with \(b_{+}\geq 2\) and a free circle action, such that \(X\) is the total space of a circle bundle over a 3-manifold \(M\) with non-torsion first Chern class \(\chi\). Let \(\xi\) be a \(\text{spin}^c\) structure over \(X\). The paper proves that if \(\xi\) is not pulled back from \(M\) then the Seiberg-Witten invariant \(SW_{X} (\xi)=0\), and if \(\xi\) is the pull back of a \(\text{spin}^c\) structure \(\xi^{*}\) over \(M\) then \(SW_{X} (\xi)\) is the sum of the Seiberg-Witten invariants \(SW_{M}(\xi')\) of the 3-manifold, over those \(\xi'\) with \(\xi' \equiv \xi^{*}\) modulo \(\chi\). This is proved by using the results of \textit{J. W. Morgan, T. S. Mrowka} and \textit{Z. Szabo} [Math. Res. Lett. 4, No. 6, 915-929 (1997; Zbl 0892.57021)]. NEWLINENEWLINENEWLINEAs an application, the author produces examples of non-symplectic 4-manifolds admitting a free circle action and whose orbit space fibers over the circle. Also he describes a non-trivial 3-manifold which is not the orbit space of any symplectic 4-manifold with a free circle action. NEWLINENEWLINENEWLINEFinally, a corollary of the main theorem gives the Seiberg-Witten invariants of the 3-manifold \(M\) which is the total space of a circle bundle over a surface.