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When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes.

Does it matter which one I use in general? Do they always yield the same homology groups?

Thank you!

Eric Wofsey
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Quoka
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    Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology. – Matematleta Dec 02 '18 at 04:40
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    @Matematleta Thanks! Then do you know why most sources use delta complexes? – Quoka Dec 02 '18 at 04:45
  • It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See [here](https://math.stackexchange.com/questions/1187445/why-delta-complex-structure) for example. – Matematleta Dec 02 '18 at 04:52
  • It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft. – Alexey Do Dec 02 '18 at 09:42

1 Answers1

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To be precise, we have a way to define homology for each different kinds of cell complexes:

  • On a simplicial complex one can define homology groups known as simplicial homology.
  • On a $\Delta$-complex one can define homology groups usually known also as simplicial homology but I will call them $\Delta$-homology to distinguish from the simplicial homology of a simplicial complex.
  • On a CW-complex one can define homology groups known as cellular homology.

So, how are these different homologies related? The answer is that they are the same in a very strong sense: not only are the homology groups isomorphic, but the chain complexes themselves are the same.

To make this precise, note first that a simplicial complex is a special case of a $\Delta$-complex and a $\Delta$-complex is a special case of a CW-complex (see my answer at Simplicial Complex vs Delta Complex vs CW Complex for more on this). Given a simplicial complex, there is a canonical isomorphism between its simplicial chain complex (whose homology is simplicial homology) and its $\Delta$-chain complex (whose homology is $\Delta$-homology). Indeed, this is completely trivial, since the definitions of the two chain complexes are pretty much exactly the same (whether they are literally the same depends on some technical details of how you choose to define them, but in any case it is immediate that they are canonically isomorphic). Similarly, given a $\Delta$-complex, its $\Delta$-chain complex is canonically isomorphic to its cellular chain complex. This is not quite as obvious since the definition of the cellular chain complex is a bit complicated but again is not difficult to verify (see my answer at If X is a simplicial complex, are the complex of simplicial chains and the one of cellular chains of X identical?).

Now, a separate and very natural question would be, what if I have a space $X$ and I (say) give it the structure of a simplicial complex in one way and the structure of a CW-complex in a different way (not the canonically way in which every simplicial complex is a CW-complex). Then will the simplicial homology be the same as the cellular homology? The answer is again yes, but this is a much deeper result. The typical way to prove this is to define a fourth homology theory called singular homology which is defined on a mere topological space with no extra structure whatsoever. You can then prove that for any CW-complex structure on a space, the cellular homology is canonically isomorphic to the singular homology (and the corresponding statements for simplicial complex or $\Delta$-complex structures then follow since they are special cases of CW-complex structures).

So, what is the point of having all these different homologies, if they're actually the same? The point is that sometimes some of them are easier to compute than others. Simplicial complexes and simplicial homology are very combinatorial which is sometimes nice for computations (and for pedagogy when you are first learning homology). On the other hand, finding a usable simplicial complex structure on a space is often much harder than finding a usable CW-complex structure, since CW-complex structures are much more flexible. $\Delta$-complexes are somewhere in the middle. Singular homology is totally horrendous to compute directly from the definition, but tends to be very useful for proving general theorems since its definition depends on no extra structure. There are also methods of computation that don't involve looking directly at any particular definition of homology but instead use high-level theorems about how the homology of different spaces are related (so, you compute the homology of a complicated space by relating it to simpler spaces).

Eric Wofsey
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