Orientation character

In algebraic topology, a branch of mathematics, an orientation character on a group [math]\displaystyle{ \pi }[/math] is a group homomorphism where:

[math]\displaystyle{ \omega\colon \pi \to \left\{\pm 1\right\} }[/math]

This notion is of particular significance in surgery theory.

Motivation

Given a manifold M, one takes [math]\displaystyle{ \pi=\pi_1 M }[/math] (the fundamental group), and then [math]\displaystyle{ \omega }[/math] sends an element of [math]\displaystyle{ \pi }[/math] to [math]\displaystyle{ -1 }[/math] if and only if the class it represents is orientation-reversing.

This map [math]\displaystyle{ \omega }[/math] 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 [math]\displaystyle{ \mathbf{Z}[\pi] }[/math], by [math]\displaystyle{ g \mapsto \omega(g)g^{-1} }[/math] (i.e., [math]\displaystyle{ \pm g^{-1} }[/math], accordingly as [math]\displaystyle{ g }[/math] is orientation preserving or reversing). This is denoted [math]\displaystyle{ \mathbf{Z}[\pi]^\omega }[/math].

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 kernel an index 2 subgroup, which determines the map completely.

References

External links

Orientation character at the Manifold Atlas

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