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I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of Lemma 1. In this lemma we appear to be using:

If $K$ is a surface in a closed orientable smooth 4-manifold $X$ that represents in homology a dual to an integral lift in homology of $w_2(X)$, then $K$ the tangent bundle of $X$ admits a trivialization along the 1-skeleton of $X - K$ that extends to the 2-skeleton of $X-K$.

I have convinced myself of this at various points, although no explanation has really taken hold. Is the converse of this true - namely, if $K$ is a surface so that $X-K$ admits a trivialization of the 1-skeleton that extends over the 2-skeleton, then is $K$ dual to an integral lift of $w_2$?

As a further question, is this characterization of $w_2$ true in an $n$-dimensional setting? It is used for 5-manifolds with boundary in the proof of Lemma 1 that I referred to above.

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    $\begingroup$ Perhaps this helps? From wikipedia: "If the manifold has a cell decomposition or a triangulation, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the tangent bundle over the 1-skeleton that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a Whitney sum with a trivial line bundle." $\endgroup$ Sep 12, 2019 at 15:24

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