CW-complexes are defined by attaching cell with increasing dimension: you start with a set of points, then attach 1-cells, then 2-cells and so on. Why are defined so? My question is: why is it necessary to attach cells ordered by dimension and not to attach a 2-cell, then a 1-cell, then a 3-cell...? The proofs I have seen about CW-complexes are mainly done inductively on the dimension of the complex, but I guess that they could be done also inductively on the number of cells attached (for example I am thinking about the uniqueness theorem for homology theories). Are there theorems that holds only for standard CW-complexes and not for these?
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11$\begingroup$ Up to homotopy these two kinds of spaces are the same, so if you only care about homotopy invariant notions there's no difference (arranging the cells by dimension is just a lot more convenient for some arguments) $\endgroup$– Denis NardinCommented Jan 15, 2020 at 9:36
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$\begingroup$ @DenisNardin Thanks, that's what I was looking for. If you post this as answer, I'll accept it. $\endgroup$– CNS709Commented Jan 15, 2020 at 10:27
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$\begingroup$ The words "convenient for some arguments" apply often to the construction of continuous functions and homotopies on a CW-complex. $\endgroup$– Ronnie BrownCommented Jan 15, 2020 at 11:46
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3$\begingroup$ The key idea is the cellular approximation theorem. If you attach a cell of lower dimension, you can make the attaching map cellular and then it is easy to see that you could also have attached the cells in the different order. Changing the attaching maps up to homotopy does not change the homotopy type. $\endgroup$– HenrikRüpingCommented Jan 15, 2020 at 12:16
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There is a name for the kind of space you are describing: a cell complex. A CW complex is a cell complex which has cell attachments in the increasing order of dimension.
The main advantage of having a CW complex over a mere cell complex is that the filtration by skeleta defines a finite chain complex model for its singular homology: if $X$ is a CW complex with $k$-skeleton $X^{(k)}$ then the homology of the pair $H_\ast(X^{(k)},X^{(k-1)})$ is concentrated in degree $k$ and is free abelian in that degree. The composition $$ H_k(X^{(k)},X^{(k-1)}) \overset{\partial}\to H_{k-1}(X^{(k-1)}) \to H_{k-1}(X^{(k-1)},X^{(k-2)}) $$ defines the boundary operator for cellular homology.
Howeover, in the cell complex case, there is no such filtration giving rise to a finite chain complex. The best thing you can do is use the partial ordering of cell attachements to define a filtration indexed by a poset whose relative homology defines a spectral sequence. This is more cumbersome to work with.
There is a thirty year old work by Igusa and Waldhausen which studies families of cell complexes--defining a sort of moduli space called the expansion space. The work was never published. Roughly speaking, they prove that the moduli space of cell complexes which are pointed and contractible give a model for the $h$-cobordism space of a disk.
