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My professor has defined the connected sum of two oriented closed connected triangulated manifolds $M_{1}$ and $M_{2}$ as $N_{1} \cup_{f} N_{2}$ where:

$N_{i}$ is the n-manifold with boundary obtained by erasing the interior of an n-cell to $M_{i}$.

$f$ is an orientation reversing homeomorphism from $\partial(N_{1})\simeq S^{n-1}$ to $\partial(N_{2})\simeq S^{n-1}$.

I have read that in that way the connected sum is also oriented, but I do not understand why it is necessary for $f$ to be orientation reversing.

Thanks in advance.

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Visualize the special case where $M_1, M_2$ are two copies of the torus $T^2$ with, say, the "clockwise" orientation. Cut out two disks from them, whose boundary circles again have the "clockwise" orientation, and now picture gluing these boundary circles together. The orientations don't match, because the boundary circles are facing each other. So when you glue them together in the obvious way the identification $f$ you use between them naturally ends up orientation-reversing.

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