Orientation character
In algebraic topology, a branch of mathematics, an orientation character on a group  is a group homomorphism where:

This notion is of particular significance in surgery theory. Motivation
Given a manifold M, one takes  (the fundamental group), and then  sends an element of  to  if and only if the class it represents is orientation-reversing. This map  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 , by  (i.e., , accordingly as  is orientation preserving or reversing). This is denoted . 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

References

External links 
Orientation character at the Manifold Atlas

Geometric topology
Group theory
Morphisms
Surgery theory