Orientation character
In algebraic topology, a branch of mathematics, an orientation character on a group is a group homomorphism to the group of two elements
where typically is the fundamental group of a manifold. This notion is of particular significance in surgery theory.
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
- In real projective spaces, the orientation character evaluates trivially on loops if the dimension is odd, and assigns -1 to noncontractible loops in even dimension.