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Let $M$ be a manifold with boundary. Hatcher writes that a compact manifold with boundary is $R$-orientable if $M - \partial M$ is $R$-orientable. That is there exists a function $x \to \mu_x \in H_n(M \vert x)$ that satisfies the local consistency property.

However, I have seen other sources talk of the orientation of the boundary induced by the orientation on the interior, i.e., the 'induced orientation'.

How do we extend the function $x \to \mu_x$ onto the boundary so to define a orientation on the boundary of a manifold?

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There are multiple ways to see this.

A differential topologist might just note that the boundary gets an orientation (of the tangent bundle) as a direct summand of the oriented tangent bundle (which can be extended to the boundary). Here you have to decide for a convention, such as inward pointing or outward pointing normal vectors (i.e. orientation of the normal line bundle).

An algebraic topologist (whose perspective you want to try to understand) might say: a consistent local orientaion will give us a fundamental class, i.e. an element $\mu \in H_n(M,\partial M)$. We also know that there is a boundary map, which would be precisely what we would like to see compatible with orientation. Hence we define the orientation of the closed manifold $\partial M$ as the image of $\mu$ under $$H_n(M,\partial M) \to H_{n-1}(\partial M).$$

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  • $\begingroup$ Of course for the fundamental class to make sense you need compactness, sorry about that. $\endgroup$ Apr 15, 2016 at 20:59

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