# Orientation of a “glued”-manifold

Im wondering if there's a short way to prove that when two manifolds with diffeomorphic boundaries are glued together along the boundaries the orientations of these must be inverse to each other. That is to say, suppose you have $M$ and $N$ oriented $n$-dimensional manifolds with $\partial M \cong \partial N$ under a diffeomorphism $\phi: \partial M \to \partial N$, you form $C = M \cup_\phi N$, in order to do this you need that the orientation of $\partial M$ be opposite to that of $\partial N$, why is that? By homological means...

I understand the reason via the orientation of the tangent spaces and the outward-first orientation of the boundaries, but how can i prove it with fundamental classes? I know the homology of the pair $(C,\partial M) \cong (M,\partial M) \oplus (N,\partial N)$ (relative Mayer-Vietoris) and the inclusion $j: (C, \emptyset) \to (C,\partial M)$ induces a monomorphism in the top homology because of the exact sequence of the pair $(C,\partial M)$, and I came up with a "proof" using this, but it is way to lengthy, maybe there is a "quick way" to do this?

Thanks

• You don't need them to be opposite. To get an orientation on the union you only need to ensure that the induced orientations on the boundary are consistent. So when you compare $M$'s induced orientation on its boundary to $N$'s, they're either the same or opposite. But you need that "sameness" or "oppositeness" to be constant from boundary component to boundary component. There is a slight preference to say "opposite" since that's what happens when the orientation on $M$ and $N$ (as submanifolds of the union) are induced by a global orientation on the union. – Ryan Budney Feb 4 '11 at 2:09
• Thank you Ryan, you're correct and that is precisely what I want to know, I guess I wrote everything wrongly. How can you see that the orientation of $C = M \cup_\phi N$ induces opposite orientations on $M$ and $N$? – Juan OS Feb 4 '11 at 2:24
• No, an orientation of $C$ does not induce "opposite" orientations of $M$ and $N$ -- those orientations are not comparable. But it does induce opposite orientations on $M \cap N$ (the common boundary of $M$ and $N$). This is due to the orientation conventions chosen for boundaries of manifolds (the "inside" direction for $M$ is the "outside" for $N$). – Ryan Budney Feb 4 '11 at 3:00
• In the above, $M \cap N$ can be oriented in two ways, as components of $\partial M$ or $\partial N$. This is the "opposite". – Ryan Budney Feb 4 '11 at 3:01
• IMO your question would be more appropriate for math.stackexchange.com, rather than here. These are issues confronted in multi-variable calculus classes and intro manifolds classes, so that's really the more appropriate forum. You can encounter them as well when studying Poincare duality in an intro algebraic topology course but I think that's not as common. Anyhow, I'd be happy to write up a long answer there. – Ryan Budney Feb 4 '11 at 3:03