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Poincaré duality

Written by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0). Orientation-unit calculation: GPT-6 Astra and GPT-6.1 Sol (OpenAI), Ultra; CC0.

On an oriented \(n\)-dimensional manifold, cohomology with compact supports in degree \(k\) is isomorphic to homology in degree \(n-k\), by capping with the orientation classes. For a closed manifold this says that cohomology and homology are mirror images of each other around the middle dimension. The proof is local-to-global: the statement is checked on \(\mathbf R^n\), and the Mayer–Vietoris sequences and direct limits of the previous lesson propagate it to every manifold. This lesson proves the theorem for every commutative coefficient ring, in particular for the finite rings \(\mathbf Z/m\), and derives the nondegeneracy of the cup product pairing over a field.

We use Orientations and fundamental classes and Cap products and cohomology with compact supports.

Basic references are [Hatcher] and [Miller].

1. The theorem

Theorem 1.1 (Poincaré duality). Let \(R\) be a commutative ring and \(M\) an \(R\)-oriented \(n\)-manifold. The duality map

\[ D_M:H^k_c(M;R)\longrightarrow H_{n-k}(M;R) \]

of Cap products and cohomology with compact supports, Section 3 is an isomorphism for every \(k\).

The proof uses three propagation properties. Say that an open subset \(U\subset M\), with the induced orientation, satisfies duality if \(D_U\) is an isomorphism in all degrees.

Lemma 1.2.

  1. If \(U\), \(V\) and \(U\cap V\) satisfy duality, so does \(U\cup V\).
  2. If \(U_1\subset U_2\subset\cdots\) all satisfy duality, so does \(\bigcup_\nu U_\nu\).
  3. An open subset of \(M\) homeomorphic to \(\mathbf R^n\) satisfies duality.

Proof. (1) By Cap products and cohomology with compact supports, Proposition 3.2, the duality maps form a map from the compact-support Mayer–Vietoris sequence of \(U,V\) to the homology Mayer–Vietoris sequence, with squares commuting up to sign. Changing the signs of some of the vertical maps does not affect whether they are isomorphisms, and after such changes the squares commute; the five lemma gives that \(D_{U\cup V}\) is an isomorphism.

(2) By Proposition 2.4 of the same lesson, \(H^k_c(\bigcup U_\nu)=\varinjlim H^k_c(U_\nu)\). Every singular chain has compact support, so it lies in some \(U_\nu\), and a cycle bounding in the union bounds in some \(U_\nu\); hence \(H_{n-k}(\bigcup U_\nu)=\varinjlim H_{n-k}(U_\nu)\). The duality maps are compatible with the inclusions by Proposition 3.1 there, and a direct limit of isomorphisms is an isomorphism.

(3) Identify the open set with \(\mathbf R^n\). Both sides vanish unless \(k=n\), by Example 2.2 of the previous lesson and contractibility. For \(k=n\), the closed balls \(\overline B\) about zero are cofinal. Choose an affine \(n\)-simplex \(\sigma\) containing \(\overline B\) in its interior. Its relative integral class generates the local homology group; the prescribed \(R\)-orientation is therefore \(\mu_{\overline B}=u[\sigma]\) for a unit \(u\in R^\times\). Indeed both classes generate the same free rank-one \(R\)-module, so the multiplier must be invertible. The integral relative chain complex is free and has homology \(\mathbf Z[\sigma]\) in degree \(n\), with zero homology in other degrees. The universal coefficient sequence therefore gives a relative cohomology generator represented by a cocycle \(\varphi\) with \(\varphi(\sigma)=1\). With the cap convention of the previous lesson,

\[ \mu_{\overline B}\frown\varphi =u\,\varphi(\sigma)[v_n]=u[v_n]. \]

Multiplication by the unit \(u\) is an automorphism of \(H_0(\mathbf R^n;R)\cong R\), so the duality map sends a generator to a generator and is an isomorphism. Neither \(u=1\) nor \(u\in\{1,-1\}\) is assumed. The cap maps commute with enlargement of the balls, so this calculation gives the required compact-support map. \(\square\)

Proof of Theorem 1.1. Open subsets of a chart. Let \(W\) be open in \(M\) and contained in a chart, identified with an open subset of \(\mathbf R^n\). A convex open subset of \(\mathbf R^n\) is homeomorphic to \(\mathbf R^n\), so it satisfies duality by Lemma 1.2(3). A finite union of convex open sets satisfies duality, by induction on the number of sets using Lemma 1.2(1): if \(U=U_1\cup\cdots\cup U_{m-1}\) and \(V=U_m\), then \(U\cap V=\bigcup_{j<m}(U_j\cap U_m)\) is a union of \(m-1\) convex open sets. Finally \(W\) is a countable union of open balls \(B_1,B_2,\ldots\), and the finite unions \(B_1\cup\cdots\cup B_m\) increase to \(W\); Lemma 1.2(2) applies.

All of \(M\). \(M\) is second countable, so it is a countable union of chart domains \(W_1,W_2,\ldots\). By induction on \(m\), \(W_1\cup\cdots\cup W_m\) satisfies duality: apply Lemma 1.2(1) with \(U=W_1\cup\cdots\cup W_{m-1}\) and \(V=W_m\), noting that \(U\cap V\) and \(V\) are open subsets of a chart. Lemma 1.2(2) concludes. \(\square\)

References: Hatcher, §3.3, develops the orientation and cap-product framework. Miller, Theorem 34.2, treats compact manifolds over a principal ideal domain; Theorem 37.1 gives the relative duality argument. The compact-support proof above treats the stated arbitrary commutative coefficient ring.

2. Closed manifolds

Corollary 2.1. If \(M\) is a closed \(R\)-oriented \(n\)-manifold with fundamental class \([M]\), then \(H^k(M;R)\to H_{n-k}(M;R)\), \(\varphi\mapsto[M]\frown\varphi\), is an isomorphism for every \(k\).

Proof. For compact \(M\), \(H^k_c(M)=H^k(M)\) and \(\mu_M=[M]\). \(\square\)

Every complex manifold is canonically oriented (Orientations and fundamental classes, Section 2), hence \(R\)-oriented for every \(R\). So Theorem 1.1 holds for every complex manifold of complex dimension \(d\) with \(n=2d\), with any coefficients, including \(\mathbf Z/m\) and \(\mathbf Z_\ell\)-modules of the form \(\mathbf Z/\ell^\nu\).

Corollary 2.2 (cup product pairing). Let \(F\) be a field and \(M\) a closed \(F\)-oriented \(n\)-manifold. For every nonzero \(\varphi\in H^k(M;F)\) there is \(\psi\in H^{n-k}(M;F)\) with \((\varphi\smile\psi)[M]\neq0\). If the cohomology of \(M\) is finite-dimensional, the pairing \((\varphi,\psi)\mapsto(\varphi\smile\psi)[M]\) is perfect.

Proof. Over a field, \(\operatorname{Hom}_F(-,F)\) is exact, so evaluation identifies \(H^j(M;F)\) with the dual of \(H_j(M;F)\). By Proposition 1.1(2) of the previous lesson, \((\varphi\smile\psi)[M]=\psi([M]\frown\varphi)\). If \(\varphi\neq0\), then \([M]\frown\varphi\neq0\) by Corollary 2.1, and some \(\psi\) is nonzero on it. With finite dimensions, the injective map \(H^k\to(H^{n-k})^*\) and the injective map \(H^{n-k}\to(H^k)^*\), obtained in the same way, force equal dimensions and bijectivity. \(\square\)

For smooth manifolds, Poincaré duality with compact supports and every commutative coefficient ring is also proved in Manifold duality, the diagonal and Wu classes, Theorem 2.2, with the Euler characteristic and Wu formula as applications.

3. Exercises

Exercise 3.1. Verify Theorem 1.1 for \(M=S^1\times\mathbf R\) with \(\mathbf Z\) coefficients.

Solution. \(M\) deformation retracts onto \(S^1\), so \(H_0=H_1=\mathbf Z\) and \(H_2=0\). Compact supports: \(M\cong\mathbf R^2\setminus\{0\}\); by the Mayer–Vietoris sequence for compact supports or directly, \(H^1_c(M)=H^2_c(M)=\mathbf Z\) and \(H^0_c=0\). Duality pairs \(H^2_c\) with \(H_0\) and \(H^1_c\) with \(H_1\).

Exercise 3.2. Show that the real projective plane is not \(\mathbf Z\)-orientable by showing that \(H_2(\mathbf{RP}^2;\mathbf Z)=0\) contradicts Corollary 2.1, and check Theorem 1.1 with \(\mathbf Z/2\) coefficients.

Solution. If it were orientable, \(H_2\cong H^0\cong\mathbf Z\). But the cellular chain complex \(\mathbf Z\xrightarrow{2}\mathbf Z\xrightarrow{0}\mathbf Z\) gives \(H_2=0\). With \(\mathbf Z/2\), the differentials vanish, so all \(H_k\) and \(H^k\), \(k=0,1,2\), are \(\mathbf Z/2\), matching \(H^k\cong H_{2-k}\).

References