= Euler chain
{title2=$\partial\xi=\sum_e(-1)^{\dim e}p_e$}
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 with
$$
\partial\xi=\sum_e(-1)^{\dim e}p_e.
$$
It 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>.<Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds by Vladimir Turaev>[Section4.1, Euler-chain definition and free transitive homology action, printed p14]{ref}
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