In learning about spin structures I was told the following: Given a manifold $X$ with a cell decomposition and a bundle $E$ over $X$, $E$ is orientable if and only if there is a trivialization of $E$ over the 0-skeleton of $X$ that extends over the 1-skeleton of $X$. Why is this true?
2026-04-04 16:18:02.1775319482
Orientation - a trivialization over the 0-skeleton that extends over the 1-skeleton
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