I had this epiphany today and i'd really like to verify it, since i'm starting to seriously doubt myself...
Let $X$ be a connected CW complex. It can be written as a colimit of it's sequence skeletons:
$$\cdots \to X^{n-1} \xrightarrow{j_{n-1}} X^n \xrightarrow{j_n} X^{n+1} \to \cdots$$
Such that at every $n \in \mathbb{N}$ there's a cofiber sequence $\bigvee S^n \xrightarrow{\phi_n} X^n \xrightarrow{j_n} X^{n+1}$.
For some reason I don't yet understand so well apart from invoking whitehead theorem (and would delight to have a simple explanatioon for) the homotopy type of a CW complex depends only on the homotopy type of the attaching maps. By this reasoning and by the cellular approximation theorem one can modify all the maps such that at every step of the sequence the attaching maps satisfy:
$$\phi_n \in [\bigvee^k S^n, \bigvee^l S^n ] \cong \mathcal{M}_{k,l}(\mathbb Z)$$
Is something wrong here?
If so can it be remedied so that a similar description is stil possible?
If not, this must make the passage to the cellular complex entirely algorithmic. Or am i missing something?
You need to consider the homotopy type of a map between a wedge of spheres and the previous stage in the CW decomposition, and there's no way to guarantee that this is another wedge of spheres of the same dimension.
For example, suppose you want to understand complexes built by starting from a wedge of $k$-spheres and attaching an $(n+1)$-cell. Then you need to understand homotopy classes of maps $S^n \to \bigvee S^k$, and these are quite complicated. In addition to needing to understand the homotopy group $\pi_n(S^k)$, you also need to take into account Whitehead brackets.