Let $M$ be a connected orientable **open** 4-manifold (noncompact, without boundary).

- Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

[**EDIT** : The answer to this is yes : see the answer of Danny Ruberman]

- Suppose now that $M$ admits a lorentzian metric
**and a spin structure**[**EDIT**]. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).