Poincaré duality

An isomorphism between the -dimensional homology groups (or modules) of an -dimensional manifold (including a generalized manifold) with coefficients in a locally constant system of groups (modules) , each isomorphic to , and the -dimensional cohomology groups of with coefficients in an orientation sheaf over (the stalk of this sheaf at the point is the local homology group ). More exactly, the usual homology groups are isomorphic to the cohomology groups , , with compact support (cohomology groups "of the second kind" ), while the homology groups "of the second kind" (determined by "infinite" chains) are isomorphic to the usual cohomology groups . In a more general form there are isomorphisms , where is any family of supports.

There are also analogous identifications between the homology and the cohomology of subsets and pairs (Poincaré–Lefschetz duality). Namely, let be an open or closed subspace in and let . Let be the family of all those sets in which are contained in and let be the family of sets of the form , . Then the exact homology sequence of the pair ,

(*)

coincides with the cohomology sequence of the pair ,

The groups coincide with when , and with when is the family of all closed sets in and the set is closed (in this case the symbol in the first sequence can be omitted, and, moreover, there is an isomorphism ). When and is open, the symbol can be omitted only in the second and third terms of the homology sequence, since the homology groups depend not only on the topological space but also on the inclusion .

When , this symbol (together with ) can be omitted in the cohomology sequence of the pair . If is closed, then

when , the cohomology of which occurs depends not only on but also on the inclusion . If and is closed, then can be replaced by and in this case is a cohomology group "of the second kind" of the space . If but is open, then the cohomology groups are not the same as (and depend on the inclusion ).

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 are isomorphic to the basic ring , then .

When the sheaf is locally constant, there exists a locally constant sheaf , unique up to an isomorphism, for which . Therefore, if in the homology sequence (*) the coefficient sheaf is used instead of , then in the cohomology sequence the sheaf appears (instead of ). 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).

References

[1]	E.G. Sklyarenko, "Homology and cohomology of general spaces" Itogi Nauk. i Tekhn. Sovr. Probl. Mat. , 50 (1989) pp. Chapt. 8
[2]	E.G. Sklyarenko, "Poincaré duality and relations between the functors Ext and Tor" Math. Notes , 28 : 5 (1980) pp. 841–845 Mat. Zametki , 28 : 5 (1980) pp. 769–776
[3]	W.S. Massey, "Homology and cohomology theory" , M. Dekker (1978)

Comments

One of the simpler forms of Poincaré duality is as follows. Let be a compact orientable manifold (cf. Orientation) and a fundamental class. Then the cap product with induces an isomorphism , 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 given by taking the wedge product of two differential forms followed by integration (with respect to a chosen volume form), giving , cf. [a3]. For Poincaré duality in the case of generalized cohomology theories defined by a spectrum , see [a4].

References

[a1]	A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Sect. VIII.8.1
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Sect. 6.2
[a3]	R. Bott, L.W. Tu, "Differential forms in algebraic topology" , Springer (1982) pp. Chapt. I, Sect. 5
[a4]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 316