For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant
$$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$
functorial with respect to quasi-isomorphisms.
Let $X$ be a manifold, and $E$ a local system on $X$, i.e., a vector bundle which can be described by constant transition functions. Fix a triangulation of $X$, and let $C^\bullet(X, E)$ be the complex computing the cohomology of $E$ using this triangulation. Then Poincaré duality gives a quasi-isomorphism
$$D_E: C^\bullet(X, E) \to C^\bullet(X,E^*)^*[-\dim X]$$
and hence a map on determinants
$$|D_E|: |C^\bullet(X, E)| \to |C^\bullet(X, E^*)|^{(-1)^{1-\dim X}}$$
On the other hand let $|E| := \Lambda^{top} E$. One also has a map
$$|D_{|E|}|: |C^\bullet(X, |E|)| \to |C^\bullet(X, |E^*|)|^{(-1)^{1-\dim X}}$$
Now the terms in the Cech complex $C^\bullet(X, |E|)$ are just the determinants of the terms in $C^\bullet(X,E)$, and the maps are correspondingly determinants, so there is a canonical isomorphism
$$|C^\bullet(X, E)| = |C^\bullet(X, |E|)|$$
$|D_{|E|}|$ and $|D_E|$ are two maps between the same one dimensional vector spaces, and so can be compared to give some number $\alpha(E)$. What is this number? In particular is it always 1?
