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Orientation character : ウィキペディア英語版
Orientation character

In algebraic topology, a branch of mathematics, an ''orientation character'' on a group \pi is a group homomorphism
:\omega\colon \pi \to \left\. 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.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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