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


1. 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]


2. 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](https://en.wikipedia.org/wiki/World_manifold) seems to say that this is not possible, but I don't understand the argument).



Open non-parallelizable 4-manifolds
Yes to your first question; the Stiefel-Whitney classes obstruct parallelizability, even for open manifolds. So for instance a non-orientable manifold (eg a Mobius band cross R^2) is not parallelizable. An oriented example would be $CP^2$ minus a point, which has nonzero $w\_2$.


I think you may have interpreted that wikipedia page incorrectly. It says that if your manifold is of the form $M^3 \times R$, with $M$ orientable, then it's parallelizable. (This follows from the fact that oriented 3-manifolds are parallelizable; you can find this in eg Milnor-Stasheff, Characteristic Classes.) But I don't think that every Lorentzian manifold is of that form; I imagine that there are some with non-trivial $w\_2$.


