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.
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