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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.^[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.^[ref]

Vladimir Turaev develops invariants in low-dimensional topology. His work on Reidemeister torsion refines invariants using Euler structures and relates them to Seiberg-Witten invariant of a three-manifold of three-manifolds.^[ref]

A formula for Dehn surgery describes how an invariant changes when a manifold is assembled by filling boundary tori. Tracking the coefficient system and the extra geometric structures is essential: the same underlying filling can have different invariant data. His paper supplies formulas in this refined setting.^[ref]

Yuhan Lim works on geometric topology and gauge-theoretic invariants of 3-manifolds. His 2000 paper gives a detailed analytic treatment of the Seiberg-Witten invariant of a three-manifold, including its dependence on the Riemannian metric and perturbation choices.^[ref]

The key issue is whether a signed count of solutions remains unchanged while those choices vary. His analysis establishes invariance for first Betti number greater than one, wall-crossing behaviour when it equals one, and the correction needed when it is zero. Thus a raw count need not itself be a topological invariant. For example, a rational homology three-sphere has first Betti number zero, placing it in the case that requires a correction. The hypotheses explain which analytic count can represent information about the underlying manifold.