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Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the resulting space is again an oriented manifold, I think I need to take an orientation reversing homeomorphism between boundaries.

So let $f:\partial M_1 \to \partial M_2$ be a homeomorphism.

My question is that:should $f$ be an orientation preserving with respect to the induced orientation on the boundaries? Or can we choose any orientations on the boundaries so that $f$ is orientation reversing with respect to the choice of the orientations of the boundaries?

Thank you.

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The resulting object will be (assuming everything's compact, say...) an orientable manifold, but not an oriented one. If you want it to be oriented, with the orientation matching that of $M_1$, say, then the map $f$ should be orientation reversing (which may not be possible without reversing the orientation on $M_2$, and hence the induced orientation on $\partial M_2$).

If you think hard about two unit disks in the plane, joined to make a sphere, you'll see what I mean.

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