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Apr 16, 2017 at 11:25 comment added R. Alexandre oh yeah I put it wrong. Maybe you have an idea about finding such a bundle over a $4$-manifold ?
Apr 16, 2017 at 11:13 comment added Mark Grant @R.Alexandre: That's right, that every orientable 4-manifold admits a Spin^c structure. I don't see why that would imply that every oriented bundle over a 4-manifold admits a Spin^c structure.
Apr 16, 2017 at 11:01 comment added R. Alexandre Thanks! I'm not very familiar with Spin (and Spin^c) structures. But as a I can see, there is no manifold in dimension less than 4 that doesn't admit a Spin^c structure. Does it mean we can't find any example in dimension less than 4?
Apr 16, 2017 at 9:49 history answered Mark Grant CC BY-SA 3.0