User:Marsupilamov/Orientation character

In algebraic topology, a branch of mathematics, an orientation character on a group $$\pi$$ is a group homomorphism
 * $$\omega\colon \pi \to \left\{\pm 1\right\}$$. This notion is of particular significance in surgery theory.

Motivation
Given a manifold M, one takes $$\pi=\pi_1 M$$ (the fundamental group), and then $$\omega$$ sends an element of $$\pi$$ to $$-1$$ if and only if the class it represents is orientation-reversing.

This map $$\omega$$ 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 (*-ring structure) on the group ring $$\mathbf{Z}[\pi]$$, by $$g \mapsto \omega(g)g^{-1}$$ (i.e., $$\pm g^{-1}$$, accordingly as $$g$$ is orientation preserving or reversing). This is denoted $$\mathbf{Z}[\pi]^\omega$$.

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.

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