I saw in the textbook affirmation that any group can be represented as the fundamental group of a $2$-dimensional topological space. Without proof. May be you can give some ideas, how can I proof that or where I can find the proof.
I know, that any group can be represent like quatient group of free group. And I know that the number of elements of generating set of group -- number of "holes" in space. But why only $2$-dimensional space in this affirmation?
Take a presentation $G = \langle S \mid R \rangle$, i.e. $G$ is the quotient of the free group $\langle S \rangle$ generated by $S$, modulo the relations $R \subset \langle S \rangle$. You can consider first the wedge sum of $S$-many circles, $X_1 = \bigvee^S S^1$. Its fundamental group is $\langle S \rangle$ (immediate application of van Kampen's theorem). Then you attach a $2$-cell for each relation in $R$, along a path that represents the given element of $\langle S \rangle = \pi_1(X_1)$. In this way you obtain a CW-complex $X = X_2$ of dimension $2$, and its fundamental group is $G$ by van Kampen's theorem (again).