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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]