Let $X$ and $Y$ be two topological spaces both with the following cell structure

2 0-cells

4 1-cells

3 2-cells

1 3-cell

Can I conclude anything about the fundamental groups of $X$ and $Y$? Are they isomorphic?


No, they don't have to. You can collapse each boundary of each cell to a single point to obtain the wedge sum of all cells (together with a single isolated $0$-cell), which are spheres of different dimension. This will give you $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$ as the fundamental group on one component. The other option is to make a disk out of two $1$-cells and one $2$-cell and again wedge product of the rest. Since we now have only two $1$-cells left then the fundamental group is $\mathbb{Z}*\mathbb{Z}$.

These examples have an isolated $0$-cell but this can be fixed again by using one of the $1$-cells to connect it to the rest of the body (so it's like a hair growing out of the center of the wedge sum). That way we reduce the number of $1$-cells in both cases by $1$ obtaining fundamental groups of $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$ and $\mathbb{Z}$ respectively.

By making all $1$-cells glued with all $2$-cells (i.e. the 2 dimensional disk) and the sole $3$-cell added via wedge sum (i.e. $D^2\vee S^3$) you obtain a cw complex with trivial fundamental group. So as you can see lots of choices...

It is hardly ever the case when predefined number of cells in each dimension determines the fundamental group of the space.


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