Top compactly supported cohomology from orientation signs ($H_c^n(M;\mathbb{Z})$)

For a connected $n$-dimensional manifold without boundary,Choose a locally finite good cover by coordinate disks, each with a local orientation. Compactly supported classes involve only finitely many of these disks at a time. The compact-support Mayer–Vietoris sequence gives one generator $u_i$ per disk and overlap relations $u_i=\pm u_j$, with the sign determined by the two local orientations. Passing to the direct limit over finite unions handles a noncompact manifold; a finite good cover of the whole manifold need not exist.

The overlap graph is connected, so all generators reduce to one up to sign. If local orientations can be chosen coherently, every relation has positive sign and the result is $\mathbb{Z}$. An orientation-reversing loop imposes $u=-u$, giving $\mathbb{Z}/2$. This is also the top-degree case of Poincare duality with the orientation local system, using compactly supported cohomology on the cohomology side.