I was reading the following post on mathoverflow: https://mathoverflow.net/questions/80081/what-are-good-examples-of-spin-manifolds in an answer there is written that
Orientability means the tangent bundle trivializes over a 1-skeleton. Dually you could think of that as saying the complement of a co-dimension 2 subcomplex has a trivial tangent bundle.
As Lee Mosher suggests in the comment section, this means that the tangent bundle of an $n$ dimensional $M$ (homeomorphic to a simplicial complex) is trivial when restricted to the $1$-skeleton of $M$.
Unfortunately, I do not manage to prove that this is equivalent to the definition of orientability that I am used to (for smooth manifolds). Namely that exists a never vanishing section of the bundle of $n$-forms over an $n$ dimensional manifold. Can someone explain it to me?
Question: Show that the above definition of orientability (in the smooth setting) is equivalent to the existence of a volume form. Also I wonder why we have that "dual" definition.
A manifold is orientable if and only if its first Stiefel-Whitney class vanishes, which is equivalent to say that this class vanishes on the $1$-skeleton.
Basically, by taking a good cover $(U_i)$ that is such that $(U_i\cap U_j)$ is connected of $M$, you can associate a $1$-Cech cocycle which takes its values in $\mathbb{Z}$, it is the sign of the determinant of the coordinate change. This $1$-cocycle is the first Stiefel-Whitney class and is trivial if the manifold is oriented. And $1$-cocycle is trivial if it is trivial on the $1$-skeleton.