Orientation character

In algebraic topology, a branch of mathematics, an orientation character on a group π 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 π = π₁M (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.

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.

See also

• Whitney characteristic class
• Local system
• Twisted Poincaré duality