Orientation character

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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

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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

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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

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* 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

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The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.

## See also

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* Characteristic class
* Local system
* Twisted Poincaré duality

## References

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This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
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## External links

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* Orientation character at the Manifold Atlas

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