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

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 $\mathbb{Z}/2$.