Orientation character


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

Motivation

Given a manifold M, one takes , and then sends an element of to if and only if the class it represents is orientation-reversing.
This map is trivial if and only if M is orientable.
The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving.

Twisted group algebra

The orientation character defines a twisted involution on the group ring, by . This is denoted.

Examples

The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.