Poincaré-Alexander duality

For an oriented manifold $M$ of dimension $n$ and a compact locally contractible subset $A$, relative duality givesFor a general compact subset the right side is interpreted as Čech cohomology; local contractibility permits the ordinary singular cohomology used above. The orientation fixes the isomorphism, and its naturality identifies restriction of an ambient fundamental class with restriction of the constant class in degree zero.

The local duality isomorphism is first obtained on coordinate balls. Excision and the Mayer–Vietoris sequence make the identifications compatible under unions; passage to neighbourhood limits yields the compact-subset formulation. On regular compact neighbourhoods the same construction is Poincare-Lefschetz duality followed by the neighbourhood limit. This explains both the relative homology pair and the cohomological degree shift.

In particular, if $M$ is connected, closed and oriented and $A$ is an embedded closed ball, the relative group is $\mathbb{Z}$ only in degree $n$. The map $H_n(M)\to H_n(M,M\setminus A)$ is an isomorphism, since it corresponds to $H^0(M)\to H^0(A)$. The long exact sequence of a pair therefore gives $H_k(M\setminus A)\cong H_k(M)$ for $k<n$ and zero top homology of the complement. This application concerns removal of the closed ball itself.