Orientation of a manifold

Summary

An orientation of an $n$-dimensional manifold is a consistent choice, at every point, of one of the two local directions of rotation, or "handedness". In algebraic topology it is a continuous choice of generator of the local homology group $H_n(M, M \setminus \set{x}) \cong \Z$ at each point $x$. For smooth manifolds it is the same as a continuous choice of orientation of every tangent space. Orientations are needed for fundamental classes, Poincaré duality with integer coefficients, and integration of differential forms.