TITLE: Minus signs in boundary operators (homology)
QUESTION [2 upvotes]: Let $ab$ be an edge (1-simplex), and $\partial_1$ be the boundary operator.
I have seen both $\partial_1(ab)=-a+b$, and also $\partial_1(ab)=a-b$ in various books.
My intuition is that both should be equivalent when it comes to defining homology, however I am not very clear why they should be equivalent.
Thanks for any enlightenment.

REPLY [2 votes]: Another way to think about this is to observe that replacing $\partial$ with $-\partial$ is equivalent to merely changing your conventions for the orientations of your 1-cycles.
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