Orientation local system
= Orientation local system
{title2=$\mathcal O_M$}
Its fiber at $x$ is $H_n(M,M\setminus\{x\};\mathbb Z)\cong\mathbb Z$. Transport changes its generator by the sign of local orientation along the path. For a connected nonorientable manifold, its degree-zero cohomology is zero, because no nonzero integer is invariant under sign reversal, whereas its degree-zero homology is $\mathbb Z/2$, the coinvariants under that reversal.