I am trying to describe what is the fundamental group of a connected CW-complex in terms of the fundamental group of it's $2$-skeleton. When we have an finite number of a attaching maps I believe I can do this but for an infinite number of them I am having some trouble.
So start with a connected CW-complex $X$. We know that there will exist a spanning tree of the $1$-skeleton $\Gamma$ , and so since $\Gamma$ is a contractible subcomplex we will have that $X/\Gamma \simeq X$, and so $X$ is homotopically equivalent to a complex with a $0$-cell.
And so from now one we can assume that $X$ is a CW-complex with a single $0$-cell. For the $1$-skeleton we will have that this will be a wedge of circles which may be infinite or finite. If it's finite we can use Van-Kampen and induction. If it's infinite I believe that we have that the $1$-skeleton will be the directed colimit of $Y_k$ where $Y_k$ is a bouquet of $k$ circles. And so using the fact that the fundamental group commutes with directed colimits we get that its fundamental group will be $\vee_{i=1}^{\infty} \mathbb{Z}$.
Now for the $2$-skeleton. We know what it's the result of attaching a $2$-cell with a map $f$. It will just be the fundamental group of the $1$-skeleton with the quotient of the normal subgroup generated by $[f]$. Now if we consider the spaces $Z_k$ which is the result of attaching the maps $k$ $2$-cells with maps $f_1,...,f_k$ to the $1$-skeleton , we will have that the $2$-skeleton will be the directed colimit of the $Z_k$, and so by commutativity of the fundamental group functor with the directed colimit we get that the fundamental group of the $2$-cell is the fundamental group of the $1$-cell quotient by the group subgroup generated by $[f_i]$ with $i={1},...,{\infty}$.
Then using induction and the same ideas we can get that the fundamental group of the $n$ skeleton with $n\geq 3$ is isomorphic to the one of the $2$ skeleton and using a compactness argument we get that the fundamental group of $X$ is isomorphic to the fundamental group of the $2$ skeleton.
Now what happens if we have an uncountable number of attaching maps ? In this case I don't think I can use directed colimits to help me and I am not sure how I can compute the fundamental group of the skeletons if we do have an uncountable number of attaching maps.
Any idea regarding this is appreciated. Thanks in advance.
Maybe an idea would be to use the Van-Kampen theorem of hatcher for any family of subspaces? We would have to be careful to the intersection of the neighborhoods we choose but I think it would be one way to do it.