Orientation character explained

\pi is a group homomorphism where:\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

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 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

• Characteristic class
• Local system
• Twisted Poincaré duality

External links

• Orientation character at the Manifold Atlas