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Most of the books and texts I read about classification problems surrounding 4-manifolds which are closed and orientable (with a occasional side-track to open orientable 4-manifolds).

This is understandable when you look at it from a physics point of view, because most of the physics I have seen trust that we live in an orientable world.

But for me as a topologist, the non-orientable ones are equally important. Only there is not much about the classification. I have read something on fake structures on the $\mathbb{R}\mathbb{P}^4$, but that's about it (and it was hidden pretty well in a conference report).

My question is, are there structured efforts been done (or going on) for the non-orientable (closed, to keep it simple) classification theory of 4-manifolds, as we see for the orientable cases (i.e. possible smooth structures on them, etc)? Can someone give me some references? I know of Markov's theorem from which it follows it is idle to get a total classification.

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You've always got the $2$-sheeted orientation covering space. I take it that this allows one to replace all non orientable manifolds with orientable ones up to minor modification, or at least very controlled modification. – Olivier Bégassat Jul 21 '11 at 19:53
Side remark: from the physics point of view it is not necessary that the manifolds are orientable. Time orientation is generally preferred, but that is a disjoint concept from orientation. I've in fact met several physicists interested in such things, one of them being Ted Jacobson. – Willie Wong Jul 21 '11 at 19:56
I can see that you always get an {non-orientable, orientable} bundle copy by looking at the 2-sheeted covering space, but I fail in seeing that every non-orientable manifold can be obtained by such a bundle (i.e. are there non-orientable manifolds, not coming from covering structures). I think you have to look at the $M^4$ with $w_1(M^4)$ not vanishing, while $M^4$ isn't obtainable from a bundle. I don't see (yet) that this is always true, or always false. – Willem Noorduin Jul 24 '11 at 14:59

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