Orientation character

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Short description: Classifier of orientation-preserving loops in a topological space

In algebraic topology, a branch of mathematics, an orientation character on a group π is a group homomorphism to the group of two elements

ω:π->{±1},

where typically π is the fundamental group of a manifold. This notion is of particular significance in surgery theory.

## Contents

* 1 Motivation
* 2 Twisted group algebra
* 3 Examples
* 4 Properties
* 5 See also
* 6 References
* 7 External links

## Motivation

Given a manifold M, one takes π=π1(M) (the fundamental group), and then ω sends an element of π to −1 if and only if the class it represents is orientation-reversing.

This map ω 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], by g↦ω(g)g−1 (i.e., ±g−1, accordingly as g is orientation preserving or reversing). This is denoted 𝐙[π]ω.

## 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 at the Manifold Atlas

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