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Let $M$ be a closed Riemannian manifold and $E$ a flat Hermitian vector bundle with unitary holonomy. On $E$-valued differential forms of degree $q$, let ${\Delta }_q$ be the nonnegative Hodge Laplacian. For its positive eigenvalues, defineusing analytic continuation to $s=0$; the prime omits zero eigenvalues. The Ray-Singer analytic torsion isThis packages the form spectra into a regularized determinant, supplying an analytic counterpart of Reidemeister torsion.^[ref]

For acyclic unitary coefficients on a closed manifold, this scalar is independent of the Riemannian metric. With nonzero cohomology, the scalar alone can depend on the metric; the full construction also tracks the norm of cohomology bases. For a circle of length $L$ with trivial coefficients, the positive eigenvalues are $(2\pi k/L{)}^2$ for $k\ne 0$ in degrees zero and one. Their zeta regularization gives $T=L$, illustrating that dependence.^[ref]

Turaev torsion refines the combinatorial side's sign and cellular-lift choices; comparing analytic and combinatorial constructions requires matching those conventions.

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]

Turaev torsion is a sign-refined form of Reidemeister torsion for manifolds and chain complexes. For a knot exterior with meridian $\mu$ and first Betti number one, its group-ring normalization satisfies

The refinement resolves sign and normalization choices present in Reidemeister torsion.

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]