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.
(the fundamental group), and then
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.
The orientation character defines a twisted involution (*-ring structure) on the group ring
is orientation preserving or reversing).
The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.
This group theory-related article is a stub.