View source for Poincaré duality

An isomorphism between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730201.png" />-dimensional homology groups (or modules) of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730202.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730203.png" /> (including a generalized manifold) with coefficients in a locally constant system of groups (modules) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730204.png" />, each isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730205.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730206.png" />-dimensional cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730207.png" /> with coefficients in an orientation sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730208.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p0730209.png" /> (the stalk of this sheaf at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302010.png" /> is the local homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302011.png" />). More exactly, the usual homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302012.png" /> are isomorphic to the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302014.png" />, with compact support (cohomology groups  "of the second kind" ), while the homology groups  "of the second kind"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302015.png" /> (determined by  "infinite"  chains) are isomorphic to the usual cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302016.png" />. In a more general form there are isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302018.png" /> is any family of supports.

There are also analogous identifications between the homology and the cohomology of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302019.png" /> and pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302020.png" /> (Poincaré–Lefschetz duality). Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302021.png" /> be an open or closed subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302022.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302024.png" /> be the family of all those sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302025.png" /> which are contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302026.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302027.png" /> be the family of sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302029.png" />. Then the exact homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302030.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302032.png" /></td> </tr></table>

coincides with the cohomology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302033.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302035.png" /></td> </tr></table>

The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302036.png" /> coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302037.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302038.png" />, and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302039.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302040.png" /> is the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302041.png" /> of all closed sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302042.png" /> and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302043.png" /> is closed (in this case the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302044.png" /> in the first sequence can be omitted, and, moreover, there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302045.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302047.png" /> is open, the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302048.png" /> can be omitted only in the second and third terms of the homology sequence, since the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302049.png" /> depend not only on the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302050.png" /> but also on the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302051.png" />.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302052.png" />, this symbol (together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302053.png" />) can be omitted in the cohomology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302054.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302055.png" /> is closed, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302056.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302057.png" />, the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302058.png" /> which occurs depends not only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302059.png" /> but also on the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302060.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302062.png" /> is closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302063.png" /> can be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302064.png" /> and in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302065.png" /> is a cohomology group  "of the second kind"  of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302067.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302068.png" /> is open, then the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302069.png" /> are not the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302070.png" /> (and depend on the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302071.png" />).

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302072.png" /> are isomorphic to the basic ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302073.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302074.png" />.

When the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302075.png" /> is locally constant, there exists a locally constant sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302076.png" />, unique up to an isomorphism, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302077.png" />. Therefore, if in the homology sequence (*) the coefficient sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302078.png" /> is used instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302079.png" />, then in the cohomology sequence the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302080.png" /> appears (instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302081.png" />). 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Sklyarenko,   "Homology and cohomology of general spaces"  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''50'''  (1989)  pp. Chapt. 8</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.S. Massey,   "Homology and cohomology theory" , M. Dekker  (1978)</TD></TR></table>



====Comments====
One of the simpler forms of Poincaré duality is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302082.png" /> be a compact orientable manifold (cf. [[Orientation|Orientation]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302083.png" /> a fundamental class. Then the cap product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302084.png" /> induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302085.png" />, cf. [[#References|[a1]]]. A formulation using the slant product with an orientation class is given in [[#References|[a2]]]. Poincaré duality (for de Rham cohomology) can also be seen as coming from the natural pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302086.png" /> given by taking the wedge product of two differential forms followed by integration (with respect to a chosen volume form), giving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302087.png" />, cf. [[#References|[a3]]]. For Poincaré duality in the case of generalized cohomology theories defined by a spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073020/p07302088.png" />, see [[#References|[a4]]].

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