Poincaré duality

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (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 four consequences: the nondegeneracy of the cup product pairing over a field, the duality between ordinary and compactly supported cohomology over \(\mathbf Z/m\), the vanishing of the cohomology of an \(n\)-manifold above degree \(n\), and the integration map on compactly supported cohomology in the top degree, with its point classes and its behaviour under open embeddings, orientation-preserving homeomorphisms and products.

We use Orientations and fundamental classes and Cap products and cohomology with compact supports. From Thom classes and Euler classes we use the universal coefficient sequence, Theorem 5.2: for a chain complex \(K\) of free abelian groups and an abelian group \(G\) there is a natural exact sequence \(0\to\operatorname{Ext}(H_{m-1}(K),G)\to H^m(\operatorname{Hom}(K,G))\to\operatorname{Hom}(H_m(K),G)\to0\), whose second map evaluates cocycles on cycles.

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 is carried by a compact set, which 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 \(\mathbf R^n\), with the orientation \(\mu\) induced from \(M\). 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 \(0\) are cofinal, and for each of them \(H^n(\mathbf R^n,\mathbf R^n\setminus\overline B)\to H^n_c(\mathbf R^n)\) is an isomorphism. Choose an affine \(n\)-simplex \(\sigma\) containing \(\overline B\) in its interior. By Orientations and fundamental classes, Lemma 1.2, the integral relative chain complex \(C_\bullet(\mathbf R^n)/C_\bullet(\mathbf R^n\setminus\overline B)\) is free with homology \(\mathbf Z[\sigma]\) in degree \(n\) and zero elsewhere, and \([\sigma]\) generates \(H_n(\mathbf R^n,\mathbf R^n\setminus\overline B;R)\); so \(\mu_{\overline B}=u[\sigma]\) for a unit \(u\in R\). The universal coefficient sequence, with \(G=R\) and \(H_{n-1}=0\), shows that \(H^n(\mathbf R^n,\mathbf R^n\setminus\overline B;R)\) is free of rank one, generated by the class of a cocycle \(\varphi\) with \(\varphi(\sigma)=1\). Then \(\mu_{\overline B}\frown\varphi=u\,\varphi(\sigma)\,\sigma|[v_n]=u\,v_n\), where \(v_n\) is the last vertex viewed as a \(0\)-simplex; this is a generator of \(H_0(\mathbf R^n;R)\cong R\). So \(D\) maps a generator to a generator. \(\square\)

Proof of Theorem 1.1. Open subsets of a chart. Let \(W\) be open in \(M\) and contained in a chart domain, 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 domain. Lemma 1.2(2) concludes. \(\square\)

Reference: this proof is [Hatcher, Theorem 3.35]; [Miller, Theorem 34.2] gives the same argument.

2. Consequences

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, Corollary 3.5), hence \(R\)-oriented for every \(R\). So Theorem 1.1 holds for every complex manifold of complex dimension \(d\), with \(n=2d\) and any coefficients, including \(\mathbf Z/m\).

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. \(\square\)

Corollary 2.3 (self-injective coefficients). Let \(\Lambda\) be a commutative ring that is injective as a module over itself, and \(M\) a \(\Lambda\)-oriented \(n\)-manifold. Then for every \(k\):

  1. evaluation \(H^k(M;\Lambda)\to\operatorname{Hom}_\Lambda(H_k(M;\Lambda),\Lambda)\) is an isomorphism;
  2. the map \(H^k(M;\Lambda)\to\operatorname{Hom}_\Lambda\bigl(H^{n-k}_c(M;\Lambda),\Lambda\bigr)\), \(\varphi\mapsto\bigl(\psi\mapsto\varphi(D_M\psi)\bigr)\), is an isomorphism; on a class \(\psi\) represented on \((M,M\setminus K)\), the value is \((\psi\smile\varphi)(\mu_K)\);
  3. if \(\Lambda=\mathbf Z/m\) and \(H^{n-k}_c(M;\Lambda)\) is finite, then \(H^k(M;\Lambda)\) is finite of the same order.

The ring \(\mathbf Z/m\) is injective over itself: Poincaré duality for curves, Lemma 3.1 proves this with Baer's criterion, which is Lemma 2.1 of Injective modules and bounded-below derived functors.

Proof. (1) The chain complex \(C_\bullet(M;\Lambda)=C_\bullet(M)\otimes\Lambda\) consists of free \(\Lambda\)-modules, and \(C^\bullet(M;\Lambda)=\operatorname{Hom}_{\mathbf Z}(C_\bullet(M),\Lambda)=\operatorname{Hom}_\Lambda(C_\bullet(M;\Lambda),\Lambda)\). Since \(\Lambda\) is injective, \(\operatorname{Hom}_\Lambda(-,\Lambda)\) is exact and commutes with taking homology. (2) Compose (1) with the isomorphism \(D_M\) of Theorem 1.1; the formula for the value is Proposition 1.1(2) of the previous lesson. (3) A finite \(\mathbf Z/m\)-module is a finite direct sum of cyclic groups \(\mathbf Z/d\) with \(d\mid m\), and \(\operatorname{Hom}(\mathbf Z/d,\mathbf Z/m)\cong\mathbf Z/d\); so the dual of a finite module has the same order. Apply (2). \(\square\)

Corollary 2.4 (dimension). Let \(M\) be an \(n\)-manifold. Then \(H_i(M;\mathbf Z)=0\) for \(i>n\) and \(H_n(M;\mathbf Z)\) is free abelian. Consequently \(H^i(M;A)=0\) for \(i>n\) and every abelian group \(A\).

Proof. A singular chain is carried by a compact set, which meets only finitely many components, so \(H_i(M)\) is the direct sum of the homology of the components. For a connected noncompact component, \(H_i=0\) for \(i\geq n\) Orientations and fundamental classes, Corollary 5.2. For a compact connected component \(N\), \(H_i(N)=0\) for \(i>n\), and \(H_n(N;\mathbf Z)\) is isomorphic to the group of locally consistent families of integral classes (Corollary 5.1 there). Such a family \(\alpha\neq0\) vanishes nowhere (Proposition 2.3(1) there), and the absolute value of \(\alpha_x\) with respect to an identification \(H_n(N,N\setminus x;\mathbf Z)\cong\mathbf Z\) is locally constant, hence a constant \(c\geq1\); then \(x\mapsto\alpha_x/c\) is an orientation, since \(\alpha_D\) is divisible by \(c\) on small compact sets with convex image, by Lemma 1.2(1) there. So \(H_n(N;\mathbf Z)\) is \(\mathbf Z\) if \(N\) is orientable (generated by \([N]\)) and \(0\) otherwise. A direct sum of free groups is free. The universal coefficient sequence for \(K=C_\bullet(M)\) gives \(H^i(M;A)=0\) for \(i>n+1\), and \(H^{n+1}(M;A)\cong\operatorname{Ext}(H_n(M;\mathbf Z),A)=0\) because \(H_n\) is free. \(\square\)

Integration and point classes. Let \(M\) be an \(R\)-oriented \(n\)-manifold. For compact \(K\subset L\), the class \(\mu_L\) restricts to \(\mu_K\), so the Kronecker pairings \(H^n(M,M\setminus K;R)\to R\), \(a\mapsto\langle a,\mu_K\rangle\), are compatible and define the integration map

\[ \int_M:H^n_c(M;R)\longrightarrow R . \]

For \(x\in M\) the group \(H^n(M,M\setminus x;R)\) is \(\operatorname{Hom}_R(H_n(M,M\setminus x;R),R)\cong R\): apply the universal coefficient sequence with \(G=R\) to the free complex \(C_\bullet(M)/C_\bullet(M\setminus x)\), whose integral homology is \(\mathbf Z\) in degree \(n\) and zero elsewhere (Lemma 1.2 of the first lesson). Let \(u_x\) be the class with \(\langle u_x,\mu_x\rangle=1\), and call its image \([x]\in H^n_c(M;R)\) the point class of \(x\).

Corollary 2.5 (integration). Let \(M\) be an \(R\)-oriented \(n\)-manifold.

  1. \(\int_M=\varepsilon\circ D_M\), where \(\varepsilon:H_0(M;R)\to R\) is the augmentation. If \(M\) is connected, \(\int_M\) is an isomorphism, and \(H^n_c(M;R)\) is free of rank one, generated by the point class of any point.
  2. \(\int_M[x]=1\) for every \(x\in M\).
  3. For an open \(U\subset M\) with the induced orientation, \(\int_U=\int_M\circ e\), where \(e:H^n_c(U;R)\to H^n_c(M;R)\) is extension by zero; \(e\) sends point classes to point classes.
  4. Let \(g:U\to V\) be a homeomorphism between open subsets of \(R\)-oriented \(n\)-manifolds with \(g_*\mu_x=\mu_{g(x)}\) for every \(x\in U\). Then \(g^*u_{g(x)}=u_x\).
  5. For an \(R\)-oriented \(k\)-manifold \(N\), with the product orientation on \(M\times N\), and compactly supported \(a\in H^n_c(M;R)\), \(b\in H^k_c(N;R)\): \(\int_{M\times N}a\times b=\int_Ma\cdot\int_Nb\).

Proof. (1) For \(a\in H^n(M,M\setminus K)\), \(D_M(a)\) is represented by \(\mu_K\frown a\), and for an \(n\)-simplex \(\sigma\) the cap product \(\sigma\frown a=a(\sigma)\,\sigma|[v_n]\) has augmentation \(a(\sigma)\). So \(\varepsilon(D_Ma)=\langle a,\mu_K\rangle\). If \(M\) is connected, \(\varepsilon\) is an isomorphism and so is \(D_M\) (Theorem 1.1); the point class then generates, by (2). (2) is the case \(K=\{x\}\). (3) The excision isomorphism \(H^n(M,M\setminus K)\cong H^n(U,U\setminus K)\) for compact \(K\subset U\) is dual to the one carrying the class \(\mu_K\) of \(U\) to that of \(M\) (proof of Proposition 3.1 of the previous lesson). (4) \(\langle g^*u_{g(x)},\mu_x\rangle=\langle u_{g(x)},g_*\mu_x\rangle=1\), after the excisions to \(U\) and \(V\). (5) Represent \(a\) on \((M,M\setminus K)\) and \(b\) on \((N,N\setminus L)\); then \(a\times b\) is represented on \((M\times N,(M\times N)\setminus(K\times L))\), and Orientations and fundamental classes, Corollary 5.3 gives \(\langle a\times b,\mu_{K\times L}\rangle=\langle a,\mu_K\rangle\langle b,\nu_L\rangle\). \(\square\)

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 \(H_2=0\). Cover \(S^1\) by open arcs \(A\), \(B\) whose intersection is the disjoint union of two arcs \(A_1\), \(A_2\), and put \(U=A\times\mathbf R\), \(V=B\times\mathbf R\). Then \(U\), \(V\), \(A_1\times\mathbf R\) and \(A_2\times\mathbf R\) are homeomorphic to \(\mathbf R^2\), so by Example 2.2 of the previous lesson their compactly supported cohomology is \(\mathbf Z\) in degree \(2\) and zero otherwise; orient them by restricting an orientation of \(M\). Extension by zero from an open subset homeomorphic to \(\mathbf R^2\) into another such set is an isomorphism on \(H^2_c\): by Proposition 3.1 of the previous lesson and Lemma 1.2(3) it corresponds to the isomorphism \(H_0(\mathbf R^2)\to H_0(\mathbf R^2)\). So the compact-support Mayer–Vietoris sequence (2.1) reads \[ 0\to H^1_c(M)\to H^2_c(U\cap V)=\mathbf Z^2\xrightarrow{\ (a,b)\mapsto(a+b,\,-a-b)\ }H^2_c(U)\oplus H^2_c(V)=\mathbf Z^2\to H^2_c(M)\to0, \] and \(H^0_c(M)=0\). The middle map has kernel and cokernel \(\mathbf Z\), so \(H^1_c(M)\cong H^2_c(M)\cong\mathbf Z\). 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 \(H^k\cong H_{2-k}\).

Exercise 3.3. Let \(C\) be a compact connected Riemann surface. Show that \(H^i(C;\mathbf Z)=0\) for \(i\geq3\) and that \(H^2(C;\mathbf Z/m)\cong\mathbf Z/m\).

Solution. \(C\) is a compact connected \(2\)-manifold, canonically oriented by Corollary 3.5 of the lesson on orientations. Corollary 2.4 gives the vanishing for \(i\geq3\). By Corollary 2.3(2) with \(\Lambda=\mathbf Z/m\) and \(k=2\), \(H^2(C;\Lambda)\) is the \(\Lambda\)-dual of \(H^0_c(C;\Lambda)=H^0(C;\Lambda)=\Lambda\), hence isomorphic to \(\Lambda\).

References