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Seiberg-Witten vanishing theorem for \(S^1\)-manifolds with fixed points. (English) Zbl 1071.57027

The author studies Seiberg-Witten invariants of 4-manifolds with smooth circle actions. In his earlier papers, he studied Seiberg-Witten invariants for manifolds with free and fixed-point-free circle actions. Let \(X\) be a simply connected 4-manifold with a smooth circle action, the quotient homeomorphic to the 3-sphere, then Fintushel proved that \(X\) is a connected sum of copies of \(S^4\), \(\mathbb{C}^2\), \(\overline{\mathbb{C}^2}\) and \(S^2 \times S^2\). Suppose \(X\) is a smooth, closed, oriented 4-manifold with \(b_2^+(X)>0\) that admits a smooth \(S^1\)-action with at least one fixed point. Then using Fintushel’s result and decomposing \(X\) into an equivariant connected sum of two 4-manifolds, both with circle actions, the author proves that \(X\) contains an essential embedded sphere of nonnegative self-intersection. From this result he shows that the Seiberg-Witten invariant of \(X\) vanishes if \(b_2^+(X)>1\), and that \(X\) is diffeomorphic to \(\mathbb{CP}^2\), an \(S^2\)-bundle over a surface, or is obtained by a sequence of blow ups from \(\mathbb{CP}^2\) or from an \(S^2\)-bundle over a surface if \(X\) is symplectic

MSC:

57R57 Applications of global analysis to structures on manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R91 Equivariant algebraic topology of manifolds
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