Minus signs in boundary operators (homology)

Question

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.

Answer 1

We have that $\ker \partial=\ker (-\partial)$ and $\mathrm{im} \partial= \mathrm{im} (-\partial)$. Therefore, homology is unaffected.

Answer 2

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.