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- . This notion is of particular significance in surgery theory.
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
- 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.
The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.
- Orientation character at the Manifold Atlas