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I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined.

There's two definitions I've seen floating around. The first is at:

http://en.wikipedia.org/wiki/Simplicial_homology

The second, at

http://www.math.wsu.edu/faculty/bkrishna/FilesMath574/S12/LecNotes/Lec16_Math574_03062012.pdf

The only difference seems to be the inclusion of a factor of (-1)

My guess is that the extra factor doesn't matter, since there is some choice in how you construct chain. In other words, the fact that you're working with a FREE abelian group over the p-simplexes of your complex, flipping the signs results in an isomorphic group.

(If that's not the case, my other guess would be that the latter only works in Z/2Z, where sign doesn't matter anyway).

Is my reasoning sound? Or am I missing something?

There's two definitions I've seen floating around. The first is at:

http://en.wikipedia.org/wiki/Simplicial_homology

The second, at

http://www.math.wsu.edu/faculty/bkrishna/FilesMath574/S12/LecNotes/Lec16_Math574_03062012.pdf

The only difference seems to be the inclusion of a factor of (-1)

^{i}inside the sums.My guess is that the extra factor doesn't matter, since there is some choice in how you construct chain. In other words, the fact that you're working with a FREE abelian group over the p-simplexes of your complex, flipping the signs results in an isomorphic group.

(If that's not the case, my other guess would be that the latter only works in Z/2Z, where sign doesn't matter anyway).

Is my reasoning sound? Or am I missing something?

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