View source for Steenrod duality

An isomorphism between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875101.png" />-dimensional [[Homology group|homology group]] of a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875102.png" /> of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875103.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875104.png" />-dimensional [[Cohomology group|cohomology group]] of the complement (the homology and cohomology groups are the reduced ones). The problem was examined by N. Steenrod [[#References|[1]]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875105.png" /> is an open or closed subpolyhedron, the same isomorphism is known as [[Alexander duality|Alexander duality]], and for any open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875106.png" /> as [[Pontryagin duality|Pontryagin duality]]. The isomorphism

<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/s/s087/s087510/s0875107.png" /></td> </tr></table>

also holds for an arbitrary subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875108.png" /> (Sitnikov duality); here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s0875109.png" /> are the Steenrod–Sitnikov homology groups with compact supports, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s08751010.png" /> are the Aleksandrov–Čech cohomology groups. Alexander–Pontryagin–Steenrod–Sitnikov duality is a simple consequence of Poincaré–Lefschetz duality and of the exact sequence of a pair. It is correct not only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s08751011.png" />, but also for any manifold which is acyclic in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s08751012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087510/s08751013.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Steenrod,   "Regular cycles on compact metric spaces"  ''Ann. of Math.'' , '''41'''  (1940)  pp. 833–851</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.A. Sitnikov,   "The duality law for non-closed sets"  ''Dokl. Akad. Nauk SSSR'' , '''81'''  (1951)  pp. 359–362  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.G. Sklyarenko,   "On homology theory associated with the Aleksandrov–Čech cohomology"  ''Russian Math. Surveys'' , '''34''' :  6  (1979)  pp. 103–137  ''Uspekhi Mat. Nauk'' , '''34''' :  6  (1979)  pp. 90–118</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.S. Massey,   "Notes on homology and cohomology theory" , Yale Univ. Press  (1964)</TD></TR></table>