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For a connected finite CW complex $X$, choose a point $p_e$ in each open cell $e$. An Euler chain is a singular chain $\xi$ of degree one withIt exists exactly when the Euler characteristic of $X$ is zero. Necessity follows because the total coefficient sum of every one-chain boundary is zero, whereas the displayed zero-chain has coefficient sum $\chi (X)$.

For sufficiency, choose a base point $b$ and paths ${\gamma }_e$ from $b$ to $p_e$. Then $\xi ={\sum }_e(-1{)}^{\dim e}{\gamma }_e$ has boundary equal to the required sum minus $\chi (X)b$, so it works when $\chi (X)=0$. Connected CW complexes are path connected, which justifies choosing these paths.

With the cell points fixed, two Euler chains differ by a cycle and define the same Euler structure when that cycle is a boundary. Adding homology classes makes Euler structures a torsor for $H_1(X;\mathbb{Z})$. This extra choice removes an ambiguity in Turaev torsion.^[ref]

For a connected finite CW complex $X$ with Euler characteristic zero, choose a point $p_e$ in each cell. An Euler chain is a singular chain $\xi$ of degree one satisfyingWith these points fixed, two Euler chains define the same Euler structure when their difference is a chain boundary. Adding cycles makes their classes a set with a simply transitive group action of $H_1(X;\mathbb{Z})$. This choice fixes the cellular-lift ambiguity in Turaev torsion; a homology orientation fixes its sign.^[ref]

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]

Let $Y$ be obtained by Dehn filling a torus boundary of $X$, with core class $h\in H_1(Y)$. Choose a one-dimensional coefficient representation $\rho :H_1(Y)\to F^{\times }$ over a field, with $\rho (h)\ne 1$, and assume the associated twisted chain complexes are acyclic. Then, up to the usual torsion unit,With compatible Euler structures and homology orientations, the refined formula fixes that ambiguity. The denominator comes from the attached solid torus's circle complex. When $\rho (h)=1$, this division is unavailable; additional homology data are needed.^[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]