I've been working my way through Munkres's Topology, and came across the following question which I'm having some difficulty wrapping my head around.
Prove the following:
Theorem: If G is a finitely presented group, then there is a compact Hausdorff space $X$ whose fundamental group is isomorphic to G.
Proof: Suppose $G$ has a presentation consisting of $n$ generators and $m$ relations. Let $A$ be the wedge of $n$ circles; form an adjunction space $X$ from the union of $A$ and $m$ copies $B_1, \ldots, B_m$ of the unit ball by means of a continuous map $f: \bigcup \text{Bd } B_i \to A$.
(a) Show that $X$ is Hausdorff
(b) Prove the theorem in the case $m = 1$
(c) Proceed by induction on $m$.
I can't seem to figure out how the adjunction space $X$ is supposed to be put together. Is there an intuitive or obvious map $f$ that I'm missing? Even if one assumes that the unit balls $B_i$ are unit balls in $\mathbb{R}^2$, and thus $f$ is mapping from something that looks like $S^1$, I don't see how there can be an intuitive mapping from a union of $n$ copies of $S^1$ to a wedge sum of $m$ copies of $S^1$ when $n \neq m$. Of course when $n = m$ there's a fine geometric intuition for this construction (wedge sum of $n$ discs by a point on the boundary of each), but when not... do you just select $m$ copies of $S^1$ in $A$ to "fill in" with discs? What if $m > n$? And given that it seems like the set being constructed can be embedded in $\mathbb{R}^k$ for sufficiently high $k$, how could such a space not be Hausdorff?
Munkres notes that this is the construction of a 2-dimensional CW complex, something I wasn't familiar with prior to encountering this question. I've noted this in the title and done some additional background research to try and understand CW complexes as I've seen them around the web, but I don't think I have very good intuition for them yet. My hope is that there's a way to understand this construction in such a way that I can solve the question without needing a broader theory of CW complexes, as Munkres certainly seems to believe that such a broader theory is unnecessary.
This is a standard method to construct these spaces. Consider n circles $C_1,...,C_n$ attach them in one point the fundamental group of the resulting space $B_n$ is $F_n$ the free group generated by $n$-elements. To the circle $C_i$ corresponds the generator $a_i$ of $F_n$.
To obtain the presentation, you go recursively: suppose that you have a relation $a_1^{l_1}...a_n^{l_n}=1$. Take a 2-ball and attach its boundary to $B_n$ as follows: go around $C_1$ $l_1$ times,..., around $C_n$ $l_n$ times, the fundamental group of resulting space $X_2$ is $F_n/<a_1^{l_1}....a_n^{l_n}>$. Recursively, attach 2-balls to obtain the other relations.