Poincaré–Lefschetz duality

Remark 11.4.6. For a compact manifold with boundary, the corresponding statement is Poincaré–Lefschetz duality: \[ E_k(M)\cong E^{n-k}(M,\partial M). \] under the appropriate \(E\)-orientation hypothesis. At the spectrum level, Atiyah duality identifies the Spanier–Whitehead dual of \(M/\partial M\) with the Thom spectrum \(\th (-T_M)\). Together with the Thom isomorphism, this gives the displayed identification of the homology of \(M\) with its relative cohomology.