Steenrod duality

An isomorphism between the $p$-dimensional homology group of a compact subset $A$ of the sphere $S^n$ and the $(n-p-1)$-dimensional cohomology group of the complement (the homology and cohomology groups are the reduced ones). The problem was examined by N. Steenrod [1]. When $A$ is an open or closed subpolyhedron, the same isomorphism is known as Alexander duality, and for any open subset $A$ as 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

[1]	N. Steenrod, "Regular cycles on compact metric spaces" Ann. of Math. , 41 (1940) pp. 833–851
[2]	K.A. Sitnikov, "The duality law for non-closed sets" Dokl. Akad. Nauk SSSR , 81 (1951) pp. 359–362 (In Russian)
[3]	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
[4]	W.S. Massey, "Notes on homology and cohomology theory" , Yale Univ. Press (1964)