Orientation character

In algebraic topology, a branch of mathematics, an orientation character on a group ${\displaystyle \pi }$ is a group homomorphism to the group of two elements

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

where typically ${\displaystyle \pi }$ is the fundamental group of a manifold. This notion is of particular significance in surgery theory.

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

This map ${\displaystyle \omega }$ 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.

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

• 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.

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