Theorem 11.4.1 (Poincaré duality). Let \(E\) be a commutative ring spectrum, and let \(M\) be a closed smooth \(n\)-manifold equipped with an \(E\)-orientation of the virtual tangent bundle \([T_M]\), or equivalently of \(-T_M\) by Lemma 10.4.6. Then there is an isomorphism of spectra \[ \S [M] \otimes E \cong \hom (\S [M],E[n]). \] In particular, for every \(k \in \Z \) there is an isomorphism \[ E_k(M) \cong E^{n-k}(M). \]
Proof. By Theorem 11.3.13, the dual of the spectrum \(\S [M]\) in \(\Sp \) is \(\th (-T_M)\), and in particular we obtain an isomorphism of spectra \[ \S [M] \otimes E \cong \hom (\th (-T_M), E). \] By Lemma 10.4.6, the given orientation of \([T_M]\) induces an orientation of \(-T_M\). Since this virtual bundle has rank \(-n\), Theorem 10.4.10 gives an isomorphism \(\th (-T_M) \otimes E \cong (\S [M] \otimes E)[-n]\) in \(\Mod _E\), and it follows that \begin {align*} \hom (\th (-T_M), E) &\cong \hom _E(\th (-T_M) \otimes E, E) \\ &\cong \hom _E((\S [M] \otimes E)[-n], E) \\ &\cong \hom (\S [M][-n],E) \\ &\cong \hom (\S [M],E[n]). \end {align*}
Combining these two isomorphisms thus gives the first claim. The second claim immediately follows by applying \(\pi _k(-)\) to both sides. □
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