orientation character

{{Technical|date=June 2023}}

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

:\omega\colon \pi \to \left\{\pm 1\right\},

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

Motivation

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

This map \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 \mathbf{Z}[\pi], by g \mapsto \omega(g)g^{-1} (i.e., \pm g^{-1}, accordingly as g is orientation preserving or reversing). This is denoted \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 kernel an index 2 subgroup, which determines the map completely.

==See also==

References

{{Unreferenced|date=June 2023}}

{{Reflist}}