TITLE: What are "good" examples of spin manifolds?
QUESTION [43 upvotes]: I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:

What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (or not spin) (and why)?


For comparison, I'd consider the cylinder and the mobius strip to be "good" examples of orientable (or not) bundles.

I've read the answers to Classical geometric interpretation of spinors which are helpful, but I'd like specific examples (non-examples) to think about.

REPLY [14 votes]: If you know about Steenrod operations, here's a very convenient characterization:

A manifold $M$ is Spin iff its Poincare duality in $H^*(M,\mathbb Z/2)$ is 
  compatible with $Sq^1$ and $Sq^2$.

Similarly, oriented manifolds are those whose Poincare duality in $H^*(M,\mathbb Z/2)$ is compatible with $Sq^1$.
The story continues: String manifolds have a Poincare duality in $H^*(M,\mathbb Z/2)$ that is compatible with $Sq^1$, $Sq^2$ and $Sq^4$ (but now, that's no longer an if and only if). My paper http://arxiv.org/abs/0810.2131 with Chris Douglas and Mike Hill describes all that in detail and provides many concrete examples.