Orientation character

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In surgery theory, a branch of mathematics, an orientation character on a group $π$ is a group homomorphism

$\omega\colon \pi \to \left\{\pm 1\right\}$

1 Motivation
2 Twisted group algebra
3 Examples
4 Properties

[edit] Motivation

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

This map $ω$ 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.

[edit] 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., $\pm g$ , accordingly as $g$ is orientation preserving or reversing). This is denoted $\mathbf{Z}[\pi]^\omega$ .