= Seiberg-Witten invariant of a three-manifold
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For a closed oriented smooth <3-manifold> $Y$, the three-dimensional <Seiberg–Witten equations> involve a <unitary connection> and a <spinor field> associated with a <Spin-c structure>. A <Dirac operator> equation couples the <spinor field> to the <unitary connection>, while a curvature equation couples the <unitary connection> back to a quadratic expression in the <spinor field>. After perturbing and quotienting by <gauge transformations>, suitable signed counts, with corrections where required, define <Seiberg-Witten invariant of a three-manifold>.
The hypotheses and conventions matter. For $b_1(Y)>1$, the resulting function on <Spin-c structures> has a description by refined <Reidemeister torsion>; for $b_1(Y)=1$, chamber choices enter, and rational homology spheres require additional correction terms. For example, on $S^1\times S^2$ the first <Betti number> is one, so an unqualified count omitting the chamber data is incomplete. The invariants connect nonlinear <differential equations> with computable <topological invariants>.<Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds by Vladimir Turaev>[Introduction: first-Betti-number cases and the relation with refined torsion]{ref}
The dependence on perturbations, including wall crossing when $b_1=1$ and a correction term when $b_1=0$, is made explicit in the analytic construction.<Seiberg-Witten invariants for 3-manifolds in the case b1=0 or 1 by Yuhan Lim>[Introduction and Theorem 1]{ref}
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