= Top compactly supported cohomology from orientation signs
{title2=$H_c^n(M;\mathbb Z)$}
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 $\mathbb Z/2$.
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