Is the cell structure of a topological space unique?

Question

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?

Answer 1

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 $X$.

Answer 2

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!