View source for Steenrod duality

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An isomorphism between the $p$-dimensional [[Homology group|homology group]] of a compact subset $A$ of the sphere $S^n$ and the $(n-p-1)$-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 $A$ is an open or closed subpolyhedron, the same isomorphism is known as [[Alexander duality|Alexander duality]], and for any open subset $A$ as [[Pontryagin duality|Pontryagin duality]]. The isomorphism

$$H_p^c(A;G)=H^{n-p-1}(S^n\setminus A;G)$$

also holds for an arbitrary subset $A$ (Sitnikov duality); here the $H_p^c$ are the Steenrod–Sitnikov homology groups with compact supports, and the $H^q$ 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 $S^n$, but also for any manifold which is acyclic in dimensions $p$ and $p+1$.

====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>