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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)
##### Keywords:
Seiberg-Witten; Dirac operator; Rohlin’s invariant
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