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 [Broken] 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?