Poincaré duality for closed smooth manifolds is an immediate consequence of Atiyah duality. Recall that Poincaré duality says that when \(M\) is a closed orientable smooth \(n\)-manifold, then for each \(k \in \Z \) there is an isomorphism \[ - \cap [M] \colon H^{n-k}(M) \iso H_k(M). \]
Many readers will have encountered this statement in their algebraic topology courses, where it is usually proved in quite a down-to-earth way, based on manipulations of simplices in the singular simplicial complex of \(M\) and using a couple of reduction arguments. This proof makes it look like Poincaré duality is intrinsically a statement about singular (co)homology. However, this is not the case: there is a more general version of Poincaré duality for cohomology theories represented by commutative ring spectra.
The orientation theory of Section 10.4 applies to the virtual bundles occurring in Atiyah duality, and this is all that is needed.
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. □
The preceding proof is the abstract form of Poincaré duality. We next relate it to the classical fundamental-class formulation. The orientation of \(-T_M\) gives a Thom class \(u_{-T_M}\colon \th (-T_M) \to E[-n]\). Define the \(E\)-fundamental class \[ [M]_E \in E_n(M)=\pi _n(E\otimes \S [M]) \] to be represented, after shifting, by the composite \[ \S \xrightarrow {\coev } \S [M]\otimes \th (-T_M) \xrightarrow {\id \otimes u_{-T_M}} \S [M]\otimes E[-n]. \] Using the embedding \(M\hookrightarrow \R ^N\) chosen in the proof of Theorem 11.3.13, this is concretely the class determined by the Pontryagin–Thom collapse \(S^N\to \Th (\nu )\), followed by the diagonal \(\Delta _*\colon \Th (\nu )\to \Th (\nu )\wedge M_+\) and then the Thom class of the normal bundle on the first factor.
Proposition 11.4.2. Under the above identification, the isomorphism \(E^{n-k}(M)\to E_k(M)\) inverse to the one in Theorem 11.4.1 is cap product with the fundamental class: \[ -\cap [M]_E\colon E^{n-k}(M)\iso E_k(M). \]
Proof sketch. Represent a cohomology class by a map \(a\colon \S [M]\to E[n-k]\). Cap product with \([M]_E\) is represented, after shifting, by the composite \[ \S \xrightarrow {[M]_E} \S [M]\otimes E[-n] \xrightarrow {\S [\Delta ]\otimes \id } \S [M]\otimes \S [M]\otimes E[-n] \xrightarrow {\id \otimes a\otimes \id } \S [M]\otimes E[n-k]\otimes E[-n] \xrightarrow {\id \otimes \mu } \S [M]\otimes E[-k]. \] Substituting the definition of \([M]_E\) expresses this as the composite obtained from the Atiyah coevaluation, the class \(a\), and the Thom class. The triangle identity for the Atiyah duality datum then identifies it with the inverse of the isomorphism in Theorem 11.4.1. □
Example 11.4.3. For \(E=H\F _2\), every real vector bundle is \(E\)-orientable because \(\Line _{H\F _2}\simeq *\). Thus every closed smooth manifold satisfies mod-\(2\) Poincaré duality. For \(E=H\Z \), Example 10.4.8 identifies an \(H\Z \)-orientation with a classical orientation, or equivalently with a nullhomotopy of \(w_1(T_M)\). This recovers integral Poincaré duality for orientable manifolds.
Remark 11.4.4. A \(\mathrm {Spin}^c\)-structure on \(M\) determines a \(\KU \)-orientation of \([T_M]\) through the construction of Atiyah et al. (1964). At the other extreme, Example 10.4.9 shows that every real vector bundle is canonically \(\MO \)-orientable. This ties generalized Poincaré duality directly to the bordism theory of Section 10.3.
Example 11.4.5. The manifold \(\R P^2\) is not integrally orientable, and integral Poincaré duality indeed fails: \(H_0(\R P^2;\Z )\cong \Z \), whereas \(H^2(\R P^2;\Z )\cong \Z /2\). With \(\F _2\)-coefficients, it is orientable and Poincaré duality holds.
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.
Exercises
Exercise 11.1. Let \(C\) be the 1-category \(\Vect _k\) of vector spaces over some field \(k\). Show that every finite-dimensional vector space \(V\) is dualizable in \(\Vect _k\), with dual given by \(V^{\vee } = V^* := \Hom _k(V,k)\).
Exercise 11.2 (The dual of a quotient module). Let \(R\) be a commutative ring spectrum, let \(x\in \pi _0(R)\), and let \(R/x\) be the cofiber of multiplication by \(x\) on \(R\). The \(R\)-module \(R/x\) is perfect and hence dualizable. Compute its \(R\)-linear dual, showing that \[ \hom _R(R/x,R)\cong (R/x)[-1]. \]
Exercise 11.3. Let \(C\) be a symmetric monoidal \(\infty \)-category, and assume that \(C\) is stable and that the tensor product \(X \otimes -\colon C \to C\) is exact for every object \(X\) of \(C\).
- (1)
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Show that the zero object is dualizable.
- (2)
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Deduce that the subcategory of \(C\) consisting of the dualizable objects is closed under pushouts.
- (3)
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Apply the preceding two parts to \(C=\Sp \) to give a direct proof of Corollary 11.3.3.
Exercise 11.4 (Duals of finite wedges of spheres). Show that \(D(\S [n])\cong \S [-n]\) for every \(n\in \Z \). Deduce that \[ D\left (\bigoplus _{i=1}^r\S [n_i]\right ) \cong \bigoplus _{i=1}^r\S [-n_i]. \] Explain why the analogous statement for an infinite wedge requires replacing the wedge by a product and does not assert dualizability.
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