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The following is a question from Hatcher's "Algebraic Topology":

Let $M$ is a compact $R$-orientable n manifold, then the boundary map $\partial : H_n(M,\partial M;R) \to H_{n-1} (\partial M)$ sends a fundamental class for $(M,\partial M)$ to a fundamental class for $\partial M$.

Hatcher, however, doesn't treat the fundamental class for a relative homology. I don't know what the fundamental class for $\\(M,\partial M)$ means. Actually, I even have no idea how to define an orientation for a relative homology. Therefore I can't start over. Could you help me?

user63310
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2 Answers2

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...when $M$ is $R$-orientable, Lemma 3.27 gives a relative fundamental class $[M]$ in $H_n(M,\partial M; R)$ restricting to a given orientation at each point of $M-\partial M$.

(Hatcher, Section 3.3, Subsection «Other Forms of Duality»)

Grigory M
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To provide a little bit of intuition, one way of thinking of the fundamental class of an $n$-dimensional compact manifold $M$ (perhaps with boundary) is to triangulate $M$ and take the sum of all the $n$-dimensional faces, "compatibly oriented." In the case of a manifold without boundary, this defines a cycle. In the case of a manifold with boundary, this defined a relative cycle; i.e., the boundary of this chain is contained in $\partial M$. In fact, that $(n-1)$-chain in $\partial M$ is the orientation class for $\partial M$. Try it with a disk in $\mathbb R^2$ or $\mathbb R^3$.

Ted Shifrin
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