= Top compactly supported cohomology from orientation signs
{title2=$H_c^n(M;\mathbb Z)$}
For a connected $n$-dimensional <manifold> without boundary,
$$
H_c^n(M;\mathbb Z)\cong\begin{cases}\mathbb Z,&M\text{ orientable},\\\mathbb Z/2,&M\text{ nonorientable}.\end{cases}
$$
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.
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