 Orientation character

In algebraic topology, a branch of mathematics, an "orientation character" on a group formula_1 is a group homomorphism

Given a manifold "M", one takes formula_3 (the fundamental group), and then formula_4 sends an element of formula_1 to formula_6 if and only if the class it represents is orientation-reversing.This map formula_4 is trivial if and only if "M" is orientable.