Euler structure

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 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]