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I am reading Allen Hatcher's book. The book introduces CW complex and ∆-complex, which I am not sure whether I really understand. There are many ways to give a topological space ∆-complex structure. What arises naturally is that whether the simplicial homology groups of the space is independent of the choice of ∆-complex structures. So in theorem 2.27, Allen proves that the simplicial and singular homology groups of a space are isomorphic and therefore answers the question. But in the proof of the theorem, he uses a cell structure of the space. So I am not sure whether the homology groups of the space is independent of its cell structure.

More specifically, in lemma 2.34, it writes If X is a CW complex, then $H_k(X^{n},X^{n-1})$ is zero for k ≠ n and is free abelian for k = n, with a basis in one to one correspondence with the n cells for X. This cellular homology is useful for computation. For example, consider the case of a torus. The book gives a cell structure like this enter image description here so that $H_2(X^{2},X^{1})≅ \mathbb{Z}$. Could we add one more 1-cell to the structure so that the torus becomes?enter image description here

But then the homology group $H_2(X^{2},X^{1})$ would be different. So what is the criteria to give a cell structure of a space?

I should focus on my questions:

  1. Is the cell structure of a topolocal space unique?
  2. If it is not, is the homology groups of the space independent of the choice of the cell structure?
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  • $\begingroup$ My answer answers the question in your title. Honestly, I cannot tell what exactly the body of your question is asking! You should try to distill a clearer question out of it. $\endgroup$ – Mariano Suárez-Álvarez Feb 26 '16 at 4:05
  • $\begingroup$ I don't know why you're asking about Lemma 2.34 or a cell structure in the context of Theorem 2.27. Lemma 2.34 is not used in the proof of Theorem 2.27 (which concluded about 7 pages earlier), nor is any cell structure other than the $\Delta$-complex structure being talked about. $\endgroup$ – Eric Wofsey Feb 26 '16 at 4:16
  • $\begingroup$ Lemma 2.34 is not used in the proof of Theorem 2.27. But $X^{n}$ depends on the cell structure, right? So is the homology groups of the space independent of the choice of the cell structure?What I don't understand is that the proof uses cell structure of the space which I don't whether the homology groups are independent of. $\endgroup$ – Deepleeqe Feb 26 '16 at 4:24
  • $\begingroup$ The definition of singular homology does not involve the cell structure in any way. Of course you have to use the $\Delta$-complex structure in the proof of Theorem 2.27, since otherwise you can't even define what "simplicial homoogy" means. $\endgroup$ – Eric Wofsey Feb 26 '16 at 4:33
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The homology of a space $X$ does not depend on a choice of cell structure (at least, up to canonical isomorphism). But in your example, $H_2(X^2,X^1)$ is not the homology of $X$: it is the homology of the pair $(X^2,X^1)$. Obviously, this depends on what the spaces $X^2$ and $X^1$ are, i.e. on what cell structure you have on $X$.

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The cell structure is not at all unique.

Just consider a square and the infinitely many ways in which you can cut it in pieces!

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  • $\begingroup$ So the cellular homology groups depend on the cell structure, right?For a space, it may have different cellular homology groups, right? $\endgroup$ – Deepleeqe Feb 26 '16 at 4:41
  • $\begingroup$ @Deepleeqe: No, cellular homology is always canonically isomorphic to singular homology; see Theorem 2.35 in Hatcher. $\endgroup$ – Eric Wofsey Feb 26 '16 at 4:42
  • $\begingroup$ Sorry for my mistake. I should say the cellular chain complex depends of the cell structure, right? $\endgroup$ – Deepleeqe Feb 26 '16 at 4:45
  • $\begingroup$ @Deepleeqe: Right, the cellular chain complex depends on the cell structure, but its homology does not. $\endgroup$ – Eric Wofsey Feb 26 '16 at 4:55

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