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Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant. (English) Zbl 1238.57028
Let \(X\) be a smooth oriented homology oriented \(4\)-manifold with the integral homology of \(S^1\times S^3\). Define \( \lambda_{SW}(X)= \sharp \mathcal{M}(X, g, \beta)-\omega(X, g, \beta), \) where \(\sharp \mathcal{M}(X, g, \beta)\) is the signed count of points in the Seiberg-Witten moduli space \(\mathcal{M}(X, g, \beta)\) and \(\omega(X, g, \beta)\) is the index theoretic correction term. The authors prove that \(\lambda_{SW}(X)\) is independent of the choice of metric \(g\) and generic perturbation \(\beta\) and the reduction of \(\lambda_{SW}(X)\) modulo 2 is the Rohlin invariant of \(X\).

MSC:
57R57 Applications of global analysis to structures on manifolds
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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