Poincaré duality

An isomorphism between the $ p $-dimensional homology groups (or modules) of an $ n $-dimensional manifold $ M $ (including a generalized manifold) with coefficients in a locally constant system of groups (modules) $ {\mathcal G} $, each isomorphic to $ G $, and the $ ( n - p ) $-dimensional cohomology groups of $ M $ with coefficients in an orientation sheaf $ {\mathcal H} _ {n} ( {\mathcal G} ) $ over $ M $ (the stalk of this sheaf at the point $ x \in M $ is the local homology group $ H _ {n} ^ {x} = H _ {n} ( M , M \setminus x; {\mathcal G} ) $). More exactly, the usual homology groups $ H _ {p} ^ {c} ( M ; {\mathcal G} ) $ are isomorphic to the cohomology groups $ H _ {c} ^ {q} ( M ; {\mathcal H} _ {n} ( {\mathcal G} )) $, $ q = n - p $, with compact support (cohomology groups "of the second kind" ), while the homology groups "of the second kind" $ H _ {p} ( M ; {\mathcal G} ) $ (determined by "infinite" chains) are isomorphic to the usual cohomology groups $ H ^ {q} ( M ; {\mathcal H} _ {n} ( {\mathcal G} )) $. In a more general form there are isomorphisms $ H _ {p} ^ \Phi ( M ; {\mathcal G} ) = H _ \Phi ^ {q} ( M ; {\mathcal H} _ {n} ( {\mathcal G} )) $, where $ \Phi $ is any family of supports.

There are also analogous identifications between the homology and the cohomology of subsets $ A \subset M $ and pairs $ ( M , A ) $ (Poincaré–Lefschetz duality). Namely, let $ A $ be an open or closed subspace in $ M $ and let $ B = M \setminus A $. Let $ \Phi \mid B $ be the family of all those sets in $ \Phi $ which are contained in $ B $ and let $ \Phi \cap A $ be the family of sets of the form $ F \cap A $, $ F \in \Phi $. Then the exact homology sequence of the pair $ ( M , B ) $,

$$ \tag{* } \dots \rightarrow H _ {p} ^ {\Phi \mid B } ( B ; {\mathcal G} ) \ \rightarrow H _ {p} ^ \Phi ( M ; {\mathcal G} ) \rightarrow \ H _ {p} ^ \phi ( M ; B ; {\mathcal G} ) \rightarrow $$

$$ \rightarrow \ H _ {p-1} ^ {\Phi \mid B } ( B ; {\mathcal G} ) \rightarrow \dots , $$

coincides with the cohomology sequence of the pair $ ( M , A ) $,

$$ \dots \rightarrow H _ \Phi ^ {q} ( M , A ; {\mathcal H} _ {n} ( {\mathcal G} )) \rightarrow \ H _ \Phi ^ {q} ( M ; {\mathcal H} _ {n} ( {\mathcal G} )) \rightarrow $$

$$ \rightarrow \ H _ {\Phi \cap A } ^ {q} ( A ; {\mathcal H} _ {n} ( {\mathcal G} )) \rightarrow \ H _ \Phi ^ {q+1} ( M ; A ; {\mathcal H} _ {n} ( {\mathcal G} )) \rightarrow \dots . $$

The groups $ H _ {p} ^ {\Phi \mid B } ( B ; {\mathcal G} ) = H _ {p} ^ {\Phi \mid B } ( M ; {\mathcal G} ) $ coincide with $ H _ {p} ^ {c} ( B ; {\mathcal G} ) $ when $ \Phi = c $, and with $ H _ {p} ( B ; {\mathcal G} ) $ when $ \Phi $ is the family $ \Psi $ of all closed sets in $ M $ and the set $ B $ is closed (in this case the symbol $ \Phi $ in the first sequence can be omitted, and, moreover, there is an isomorphism $ H _ {p} ( M , B ; {\mathcal G} ) = H _ {p} ( A ; {\mathcal G} ) $). When $ \Phi = \Psi $ and $ B $ is open, the symbol $ \Phi $ can be omitted only in the second and third terms of the homology sequence, since the homology groups $ H _ {p} ^ {\Phi \mid B } ( B ; {\mathcal G} ) $ depend not only on the topological space $ B $ but also on the inclusion $ B \subset M $.

When $ \Phi = \Psi $, this symbol (together with $ \Phi \cap A $) can be omitted in the cohomology sequence of the pair $ ( M , A ) $. If $ A $ is closed, then

$$ H _ \Phi ^ {q} ( M , A ; {\mathcal H} _ {n} ( {\mathcal G} ) ) = \ H _ {\Phi \mid B } ^ {q} ( M ; {\mathcal H} _ {n} ( {\mathcal G} )) \ = H _ {\Phi \mid B } ^ {q} ( B ; {\mathcal H} _ {n} ( {\mathcal G} )) ; $$

when $ \Phi = \Psi $, the cohomology of $ B $ which occurs depends not only on $ B $ but also on the inclusion $ B \subset M $. If $ \Phi = c $ and $ A $ is closed, then $ \Phi \cap A $ can be replaced by $ c $ and in this case $ H _ {c} ^ {q} ( M ; A ; {\mathcal H} _ {n} ( {\mathcal G} ) ) = H _ {c} ^ {q} ( B ; {\mathcal H} _ {n} ( {\mathcal G} ) ) $ is a cohomology group "of the second kind" of the space $ B $. If $ \Phi = c $ but $ A $ is open, then the cohomology groups $ H _ {c \cap A } ^ {q} ( A ; {\mathcal H} _ {n} ( {\mathcal G} )) $ are not the same as $ H _ {c} ^ {q} ( A ; {\mathcal H} _ {n} ( {\mathcal G} )) $ (and depend on the inclusion $ A \subset M $).

Poincaré–Lefschetz duality can easily be applied to describe a duality between the homology and the cohomology of a manifold with boundary. It is useful to bear in mind that, if all the non-zero stalks of the sheaf $ {\mathcal H} _ {n} ( R) $ are isomorphic to the basic ring $ R $, then $ {\mathcal H} _ {n} ( {\mathcal G} ) = {\mathcal H} _ {n} ( R) \otimes _ {R} {\mathcal G} $.

When the sheaf $ {\mathcal H} _ {n} ( R) $ is locally constant, there exists a locally constant sheaf $ {\mathcal L} ( R) $, unique up to an isomorphism, for which $ {\mathcal L} ( R) \otimes _ {R} {\mathcal H} _ {n} ( R) = R $. Therefore, if in the homology sequence (*) the coefficient sheaf $ {\mathcal L} ( R) \otimes _ {R} {\mathcal G} $ is used instead of $ {\mathcal G} $, then in the cohomology sequence the sheaf $ {\mathcal G} $ appears (instead of $ {\mathcal H} _ {n} ( {\mathcal G} ) $). Thus, the pre-assigned coefficients can appear in the duality isomorphism either in the homology or in the cohomology.

The most natural proof of Poincaré duality is obtained by means of sheaf theory. Poincaré duality in topology is a particular case of Poincaré-type duality relations which are true for derived functors in homological algebra (another particular case is Poincaré-type duality for homology and cohomology of groups).

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One of the simpler forms of Poincaré duality is as follows. Let $ M ^ {n} $ be a compact orientable manifold (cf. Orientation) and $ c _ {M} \in H _ {n} ( M; \mathbf Z ) $ a fundamental class. Then the cap product with $ c _ {M} $ induces an isomorphism $ H ^ {i} ( M ; G) \rightarrow H _ {n-i} ( M ; G) $, cf. [a1]. A formulation using the slant product with an orientation class is given in [a2]. Poincaré duality (for de Rham cohomology) can also be seen as coming from the natural pairing $ H ^ {q} ( M) \otimes H _ {c} ^ {n-q} ( M) \rightarrow \mathbf R $ given by taking the wedge product of two differential forms followed by integration (with respect to a chosen volume form), giving $ H ^ {q} ( M) \simeq H _ {c} ^ {n-q} ( M) ^ {*} $, cf. [a3]. For Poincaré duality in the case of generalized cohomology theories defined by a spectrum $ E $, see [a4].

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