I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:



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> What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (or not spin) (and why)?
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For comparison, I'd consider the cylinder and the mobius strip to be "good" examples of orientable (or not) bundles.




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I've read the answers to [Classical geometric interpretation of spinors](https://mathoverflow.net/questions/66681/classical-geometric-interpretation-of-spinors) which are helpful, but I'd like specific examples (non-examples) to think about. 



What are "good" examples of spin manifolds?
There's the traditional obstruction-theoretic perspective. 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. 


So admitting a spin structure is the same, but it will be the tangent bundle trivializes over a 2-skeleton, dually the complement of a co-dimension three subcomplex admits a trivial tangent bundle. 


A surface is orientable if and only if it contains no Moebius bands -- a regular neighbourhood of any simple closed curve must be a cylinder. In higher dimensions this translates into a manifold being orientable if and only if it contains no twisted bundles $D^{n-1} \rtimes S^1$, i.e. regular neighbourhoods of simple closed curves are diffeomorphic to $D^{n-1} \times S^1$. 


For spin structures there's something very similar. Of course, a surface admits a spin structure if and only if it is orientable. It's a more interesting notion in higher dimensions. The statement there is the manifold is orientable, and if you take a regular neighbourhood of any surface in the manifold, then it has a trivial tangent bundle. So manifolds like $\mathbb RP^3$ are perfectly valid spin manifolds -- $\mathbb RP^3$ contains $\mathbb RP^2$ but the total space of its normal bundle has a perfectly trivializable tangent bundle. Technically, the condition is a little stronger than that -- you can trivialize the tangent bundle of the complement of a co-dimension $3$ subset. So not only can you trivialize the total spaces of normal bundles of surfaces, but even the regular neighbourhoods of unions of surfaces. 


So if you want a manifold that isn't spin, the archetype would be a vector bundle over a surface so that the total space does not have a trivializable tangent bundle. Take the $D^2$-bundle over $S^2$ with Euler Class $\chi$. I think this happens if and only if $\chi$ is even. I suppose you have more entertaining examples when dealing with the regular neighbourhood of a 2-complex that isn't itself a manifold. 


edit: Milnor's "Spin structures on manifolds" in L'Enseignement Mathematique Vol 9 (1963) is an excellent reference for most of the above. I don't believe he goes into all the descriptions above since I think he wants to keep the article simple. The Poincare duality interpretation above is a very standard mode of thinking that's employed throughout much of low-dimensional topology. Kirby's book on 4-manifolds is a nice place to look for this material. Specifically, R. Kirby "The topology of 4-manifolds" Springer-Verlag (1989). A more modern reference would be Gompf and Stipsicz, but again I don't think they use all the above descriptions. Milnor and Stasheff's "Characteristic Classes" describes most of the basic constructions involved above, in the obstruction theory section. In a couple of months I'll be putting up a paper on the arXiv that gives some very combinatorial ways of describing spin and spin^c-structures on manifolds (mostly for computer implementation). I hope that will be a good reference, too! But the paper is still unreadable. 


