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I am reading a paper in which , if I understood well, a spin structure in a manifold M

is equivalent to M admitting a trivialization of the tg.bundle over the 1-skeleton of M

( I guess M is assumed to be "nice-enough" so that it is a simplicial complex )

so that this trivialization extends to a trivialization of the 2-skeleton; then , e.g., S^2 (seen as a

2-simplex, with 1-simplices as S^1's ) would not be spin, since a trivialization over any 1-subskeleton S^1

of S^2 cannot extend to a trivialization of the bundle over S^2 --because the

bundle over S^2 is not trivial, . A spin structure (in this layout) is a choice, up to homotopy

of a (global) trivialization of the bundle over the 1-skeleton that extends to a trivialization over the

2-skeleton.

O.K, so far. But now there is a different use of spin structure which is supposedly

equivalent to the one just given: a 4-manifold is spin if every homology 2-class can be

repd. by an embedded sphere; ** BUT ** , while then S^2 is not Spin by the first

def ( bundle over S^2 is not trivial, by , e.g. the Hairy Ball Thm. ), it is spin by

the second def., since its second homology class is S^2 itself --because S^2 is

orientable. Am I missing something here somewhere?

Any Ideas?

Thanks in Advance.

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# Def./Examples of Spin Structure Re Trivial Bundles?

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