Let's say X is an orientable manifold, and X is homeomorphic to Y. Must it be that Y is also orientable?

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  • $\begingroup$ I think you have to be a little careful: your homeo space need not be a smooth manifold, so what would a smooth choice of orientation of tangent spaces mean? $\endgroup$ – Randall Nov 7 '17 at 23:35
  • $\begingroup$ @AnthonyCarapetis From your link I see it can be stated in terms of homology when the manifold is closed and connected, however it seems to be the same definition involving oriented atlases when that is not the case $\endgroup$ – Santiago Estupiñán Nov 7 '17 at 23:38
  • $\begingroup$ A manifold is orientable iff all its components are, so assuming connectedness is no loss. You're right about the compactness assumption, though - it's not as simple as just looking at the fundamental class in general. I think the general idea is still essentially true, however; the formulation in terms of homology is just more complicated. See e.g. map.mpim-bonn.mpg.de/Orientation_of_manifolds $\endgroup$ – Anthony Carapetis Nov 7 '17 at 23:54

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