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Orientation of cells

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What is orientation of a cell in a cell complex? It is an ordering of its vertices up to an even permutation.

More precisely:

There are two equivalence classes of orderings of any set, in particular on the set of vertices,

 two orderings are equivalent if they differ by an even permutation.

(Recall, even permutation = even # of transpositions.)

For example, $ABC$ isn't equivalent to $ACB$ (one flip), but it is equivalent to $CAB$ (two flips).

We assign an orientation to each cell in the complex in order to define the boundary operator. Different choices will produce different boundary operators and, therefore, different cycle groups and boundary groups (same rank/dimension though). The homology group is independent of the chosen orientations.

Orientable manifolds have cells with "compatible" orientations.