# Is the cell structure of a topological space unique?

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 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? 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?
• 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. – Mariano Suárez-Álvarez Feb 26 '16 at 4:05
• 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. – Eric Wofsey Feb 26 '16 at 4:16
• 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. – Deepleeqe Feb 26 '16 at 4:24
• 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. – Eric Wofsey Feb 26 '16 at 4:33

## 2 Answers

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 $X$.

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!

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