In algebraic topology, a branch of mathematics, an orientation character on a group π {\displaystyle \pi } is a group homomorphism to the group of two elements
ω : π → { ± 1 } {\displaystyle \omega \colon \pi \to \left\{\pm 1\right\}} , where typically π {\displaystyle \pi } is the fundamental group of a manifold. This notion is of particular significance in surgery theory.
Motivation Given a manifold M, one takes π = π 1 ( M ) {\displaystyle \pi =\pi _{1}(M)} (the fundamental group), and then ω {\displaystyle \omega } sends an element of π {\displaystyle \pi } to − 1 {\displaystyle -1} if and only if the class it represents is orientation-reversing. This map ω {\displaystyle \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 The orientation character defines a twisted involution (*-ring structure) on the group ring Z [ π ] {\displaystyle \mathbf {Z} [\pi ]} , by g ↦ ω ( g ) g − 1 {\displaystyle g\mapsto \omega (g)g^{-1}} (i.e., ± g − 1 {\displaystyle \pm g^{-1}} , accordingly as g {\displaystyle g} is orientation preserving or reversing). This is denoted Z [ π ] ω {\displaystyle \mathbf {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 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 Archived 2014-04-23 at the Wayback Machine at the Manifold Atlas
