Editing Geometric topology (section)

===Orientability===
{{main|Orientability}}
A manifold is orientable if it has a consistent choice of [[orientation (mathematics)|orientation]], and a [[connected space|connected]] orientable manifold has exactly two different possible orientations.  In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality.  Formulations applicable to general topological manifolds often employ methods of [[homology theory]], whereas for [[differentiable manifolds]] more structure is present, allowing a formulation in terms of [[differential form]]s.  An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a [[fiber bundle]]) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.