Duality Galore

On an orientable \(n\)-dimensional manifold \(M\), the following four spaces are all isomorphic: \[H_i(M;\RR), H_{n-i}(M;\RR), H^i(M;\RR), H^{n-i}(M;\RR).\] This fact really boils down to two distinct isomorphisms: \(H_i(M;\RR) \simeq H^i(M;\RR)\) and \(H_i(M;\RR) \simeq H^{n-i}(M;\RR)\). The first (known as the universal coefficients theorem), behaves surprisingly differently from the second (known as Poincaré duality) . This can be especially confusing on two-dimensional surfaces where \(H^1(M;\RR)\) is literally the same space as \(H^{2-1}(M;\RR)\). But, even then the two isomorphisms are quite different! The first isomorphism follows fairly directly from the definition of cohomology, whereas the second is a deep result about the structure of manifolds.

The first isomorphism

Recall that in (simplicial) homology, our fundamental objects of study are simplicial chains, i.e. formal linear combinations of simplices. We let \(C_k(\RR)\) denote the set of \(k\)-chains (i.e. combinations of \(k\)-dimensional simplices with real coefficients). The boundary operator \(\partial_k : C_k(\RR) \to C_{k-1}\) takes a \(k\) dimensional simplex to its \(k-1\)-dimensional boundary. Together, all of these fit together into a chain complex: \[ \ldots \xrightarrow{\partial_{k+2}}C_{k+1}(\RR)\xrightarrow{\partial_{k+1}} C_k(\RR) \xrightarrow{\partial_k} C_{k-1}(\RR) \xrightarrow{\partial_{k-1}} \ldots\] The \(i\)th homology group \(H_i(M;\RR)\) is defined to be \(\ker \partial_k / \im \partial_{k+1}\). Concretely, an element of \(H_i(M;\RR)\) may be represented by a closed chain \(c_k\), and two such representatives are homologous if they differ by a boundary \(\partial_{k+1}A\).

To define cohomology, we dualize the whole picture. We let \(C^k(\RR)\) denote the space of \(k\)-cochains, i.e. the dual space of \(C_k(\RR)\). Concretely, a \(k\)-cochain still looks like an assignment of a number to each \(k\)-simplex, but it is helpful to distinguish the two. We define \(d_k : C^k(\RR) \to C^{k+1}(\RR)\) as the adjoint of \(\partial_{k+1}\). These fit into a chain complex going the other way: \[ \ldots \xleftarrow{d_{k+1}}C^{k+1}(\RR)\xleftarrow{d_{k}} C^k(\RR) \xleftarrow{d_{k-1}} C^{k-1}(\RR) \xleftarrow{d_{k-2}} \ldots\] Now, we define the \(i\)th cohomology group \(H^i(M;\RR)\) as \(\ker d_k / \im d_{k-1}\). Concretely, an element of \(H^i(M;\RR)\) may be represented by a closed cochain \(\gamma_k\), and two such representatives are cohomologous if they differ by a coboundary \(d_{k-1}\alpha\).

Since we dualized at the level of chains, rather than directly dualizing the homology groups, it's not immediately obvious that \(H^i\) should be dual to \(H_i\). But it follows very quickly from the definitions. We just need to check that the pairing between \(C_k(\RR)\) and \(C^k(\RR)\) descends to (co)homology classes. Suppose we have a closed chain \(c\) and closed cochain \(\gamma\). Then \[ \begin{aligned} \pair{c}{\gamma + d\alpha} &= \pair c \gamma + \pair{c}{d\alpha},\\ &= \pair c \gamma + \pair{\partial c}{\alpha},\\ &= \pair c \gamma, \end{aligned} \] using the fact that \(d\) is the adjoint of \(\partial\) and \(c\) is closed. Hence, the pairing is well-defined on cohomology classes. An analogous calculation shows that it is well-defined on homology classes as well.

Furthermore, it turns out that (on nice space, using real coefficients) this pairing is nondegenerate. The existence of a nondegenerate pairing identifies \(H^i(M;\RR)\) as the dual space of \(H_i(M;\RR)\). Since these are finite-dimensional vector spaces, we can conclude that they are isomorphic. Note, though, that this isomorphism is generally not canonical (i.e. the two spaces are not naturally isomorphic). One must pick a basis of \(H_i(M;\RR)\) to map between the two, and the resulting map depends entirely on the chosen basis.

So far, we haven't used the manifold structure of \(M\) at all! And indeed, the universal coefficient theorem applies to all topological space, although things get slightly more complicated if one wants to use integer coefficients rather than the real coefficients that we've been using.

Poincaré Duality

Our second isomorphism, though, is deeply entwined with the manifold structure of \(M\). Indeed, even stating that \(H_i(M;\RR) \simeq H^{n-i}(M;\RR)\) requires the dimension of \(M\), which doesn't necessarily make sense on general topological spaces.

When working with real coefficients, it can be illuminating to write the isomorphism as \((H^i(M;\RR))^* \simeq H^{n-i}(M;\RR)\) instead (which is equivalent thanks to our first isomorphsim). In this form, Poincaré duality asserts the existence of a nondegenerate pairing between \(H^i(M;\RR)\) and \(H^{n-i}(M;\RR)\).

In de Rham cohomology this pairing is given by the wedge product: \(\pair{[\alpha]}{[\beta]} \mapsto \int_M \alpha \wedge \beta\). The more traditional Poincaré duality map \(H_i(M;\RR) \to H^{n-i}(M;\RR)\) essentially pulls back this pairing along the isomorphism \(H_i(M;\RR) \to (H^i(M;\RR))^*\). Concretely, given a homology class \([\Gamma] \in H_i(M;\RR)\), we can view integration over \([\Gamma]\) as a linear functional on \(H_i(M;\RR)\). Since the wedge product pairing is nondegenerate, we can represent this using the wedge product (or cup product) against some form \(\gamma\): \[ \int_\Gamma \omega = \int_M \gamma \wedge \omega\;\forall \omega. \] Poincaré duality is then the mapping \(H_i(M;\RR) \to H^{n-i}(M;\RR)\) given by \([\Gamma] \mapsto [\gamma]\). Note that this mapping is canonical; it does not depend on any arbitrary choices, or even a metric on \(M\).

But if you do have a metric, you can consider another nondegenerate pairing of differential forms: the inner product. The inner product of two \(k\)-forms can be written as \[ \langle\langle\alpha, \beta\rangle\rangle = \int_M \alpha \wedge \star \beta. \] The Hodge star operator gives us a mapping from \(k\) forms to \(n-k\)-forms which we can interpret abstractly as using the metric to identify \(H^k(M;\RR)\) with its dual, and then using the Poincaré duality pairing to identify its dual with \(H^{n-k}(M;\RR)\).

So far, we've looked at Poincaré duality as a pairing on cohomology classes, and we've pulled it back along one component to get a mapping from homology classes to cohomology classes. If we pull back both components, then we get a pairing on homology classes: the intersection pairing. If \(N_1\) and \(N_2\) are submanifolds of \(M\) of complementary dimension, then the intersection pairing of \([N_1]\) and \([N_2]\) counts signed intersections between the submanfolds.

Duality on surfaces

On surfaces, both the universal coefficient theorem and Poincaré duality give maps from \(H_1(M;\RR)\) to \(H^1(M;\RR)\). But these maps are quite different! To obtain the universal coefficient theorem mapping, we first fix a basis for \(H_1(M;\RR)\). Then, we map each basis loop to a 1-form which integrates to 1 along that loop and zero along all others. This gives us a well-defined duality mapping, but it depends on the choice of basis. On the other hand, Poincaré duality maps any loop to a 1-form whose integral counts intersections with that loop. This gives a canonical mapping, not requiring a choice of basis.