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The mapping torus of an order-six rotation of a hexagonal flat torus is a closed orientable flat Riemannian manifold with holonomy $C_6$. The screw rotation has angle $\pi /3$ and axial displacement $1/6$. Its first Betti number is one.

A pair of closed orientable flat three-manifolds with equal scalar Laplace-Beltrami operator spectra but different first Betti numbers, respectively one and zero. Consequently they are not homeomorphic. They cannot be isospectral for the Hodge Laplacian on one-forms, whose zero-eigenvalue multiplicity is the first Betti number. This shows why the spectrum on functions must not be silently replaced by the spectra on all differential forms.

The mapping torus of an order-three rotation of a hexagonal flat torus is a closed orientable flat Riemannian manifold with holonomy $C_3$. A screw rotation through $2\pi /3$ with axial displacement $1/3$, together with the planar lattice, describes its deck group. Its first Betti number is one.

A homeomorphism preserves every homology group, hence the Euler characteristic wherever the finite rational Betti number sum is defined. In fact, homotopy invariance of homology makes Euler characteristic invariant under homotopy equivalence. Thus different finite CW complex structures on the same space have equal alternating cell counts.^[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.