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Let $X$ be Hausdorff. Suppose further that it is triangulable. Let $K$ and $L$ be two simplicial complexes such that their underlying space $|K|=|L|=X$. It is a lot of work to show (see Chapter 2 of Munkres' Elements of Algebraic Topology) that the simplicial homology groups are the same, i.e., independent of the simplicial structure and only dependent on the underlying space.

Compare the situation with a CW-complex. The fact that the cellular homology is independent of the filtration of the space is proved in only two pages by Hatcher as the result that $H^{CW}_n(X) \simeq H_n(X)$. The proof is not only much shorter but works for a wider category.

My question, which may seem strange, is:

Where is all the work going?

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