Compactly supported cohomology ($H_c^{\ast }(X)$)

Compactly supported cohomology is the cohomology of cochains vanishing outside a compact subset. Equivalently,

For a connected n-manifold without boundary, a finite good cover presents its top compactly supported cohomology by one generator per disc and relations $u_i=\pm u_j$ on overlaps. This follows from the compact-support Mayer–Vietoris sequence and vanishing above dimension n. The overlap graph is connected, so the group is cyclic. Coherent local orientations give $\mathbb{Z}$; an orientation-reversing loop gives $\mathbb{Z}/2$.

For a nonnegative integer $d$ and an abelian group $A$,The one-point compactification is the sphere $S^d$, and it is locally contractible at the added point. Apply the compact-support comparison with a one-point compactification and the reduced cohomology of a sphere. In dimension zero the compactification is $S^0$, giving the same formula.

If the one-point compactification $X^{+}$ is Hausdorff and has a basis of contractible neighbourhoods at the added point, thenThe Excision theorem identifies $H^{\ast }(X,X\setminus K)$ with $H^{\ast }(X^{+},X^{+}\setminus K)$. The contractible neighbourhoods form a cofinal family of the complements of compact $K$, and their pair long exact sequences identify each relative group naturally with reduced cohomology. Passing to the direct limit gives the comparison. Local contractibility matters for singular cohomology; the Hawaiian earring illustrates its failure.

A continuous map is proper when inverse images of compact sets are compact. Proper maps induce contravariant maps on compactly supported cohomology.