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I've been wondering why the boundary of a simplex $\sigma : C_q (X) \rightarrow C_{q-1}(X)$ is defined to be $\partial \sigma = \sum (-1)^i \sigma \circ f_{i,q}$ with alternating sign.

Why can it not be the sum over all faces without alternating sign? Thanks for your help!

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    All the definitions in simplicial homology were inspired by corresponding notions among manifolds. In particular, if you have a manifold equipped with an orientation, there is a canonical orientation associated to its boundary. Orientations can also be assigned to objects via explicit parametrization by other oriented objects. The formula you're interested in is a combination of these two things. I suggest reading a book on manifolds that covers the general Stokes theorem (using differential forms). Once you understand the proof of the theorem, you'll be ready to reinterpret that formula.2011-08-04

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Because we want to have $\partial\circ\partial=0$ to built homology.

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    But I meant in the definition of the boundary operator. Homology with coefficients refers to the coefficients of a chain not the coefficients in the definition of the boundary...2011-07-31