Show that if \(M\) is a compact \(R\)-orientable \(n\)-manifold, then the boundary map \(H_n(M, \partial M; R) \to H_{n-1}(\partial M; R)\) sends a fundamental class for \((M, \partial M)\) to a fundamental class for \(\partial M\).
Solution
Let \[ [M,\partial M]\in H_n(M,\partial M;R) \] be a fundamental class, and let \[ \delta:H_n(M,\partial M;R)\to H_{n-1}(\partial M;R) \] be the boundary homomorphism.
Fix \(x\in\partial M\) and choose an oriented collar half-ball neighborhood of \(x\). Naturality of the long exact sequence of a pair gives a commutative diagram from the global boundary map to the local boundary map. The image of \([M,\partial M]\) in the local relative group is the local fundamental class of the half-ball, and its local boundary is precisely the generator of \[ H_{n-1}(\partial M,\partial M-\{x\};R) \] that defines the induced boundary orientation from Exercise 30.
Therefore \(\delta[M,\partial M]\) maps to the chosen local orientation generator at every \(x\in\partial M\). A fundamental class is characterized by this property. Hence \[ \boxed{\delta[M,\partial M]=[\partial M]} \] for the induced orientation on \(\partial M\) (with the usual global sign convention).