In algebraic topology, a branch of mathematics, an orientation character on a group is a group homomorphism to the group of two elements

1. ,

is the fundamental group of a manifold. This notion is of particular significance in surgery theory.

Motivation

Given a manifold M if and only if the class it represents is orientation-reversing.

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.

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 as its kernel an index 2 subgroup, which determines the map completely.

See also

• Characteristic class

• Local system

• Twisted Poincaré duality

References

External links

• Orientation character at the Manifold Atlas