Editing Orientability (section)

===Orientation and cohomology===

A manifold ''M'' is orientable if and only if the first [[Stiefel–Whitney class]] <math>w_1(M) \in H^1(M; \mathbf{Z}/2)</math> vanishes. In particular, if the first cohomology group with '''Z'''/2 coefficients is zero, then the manifold is orientable. Moreover, if ''M'' is orientable and ''w''<sub>1</sub> vanishes, then <math>H^0(M; \mathbf{Z}/2)</math> parametrizes the choices of orientations.<ref>{{Cite book | last1=Lawson | first1=H. Blaine | author1-link=H. Blaine Lawson |  last2=Michelsohn | first2=Marie-Louise | author2-link=Marie-Louise Michelsohn | title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=0-691-08542-0 | year=1989 |page=79 Theorem 1.2}}</ref> This characterization of orientability extends to [[orientation of a vector bundle|orientability of general vector bundles]] over ''M'', not just the tangent bundle.