[Math] Hatcher 3.3 Exercise 31

algebraic-topology

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?

Best Answer

...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»)

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