In algebraic topology, a branch of mathematics, an orientation character on a group
Contents
Motivation
Given a manifold M, one takes
This map
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
Properties
The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.