Given a space $G = \bigcup_{n=1}^{\infty} A_n$ where $ A_n $ is a circle $ C[ (n,0),n] \in R^{2}$ I'd like to show that its fundamental group is an infinately generated free group.
So let's say $a_i $ denotes a loop that begins at $(0,0) $ and goes around circle $A_i$ once. Now If we are given any loop $ \alpha : I \rightarrow G $
$$\alpha (0)= (0,0) $$
Then since $I$ is compact the image of $\alpha$ may lie in finitely many circles $A_i$ ( let's say the image of alpha belongs to circles $A_{i_1},…,A_{i_k}$) and thus from Van Kampen theorem I know that $$\alpha \in <a_{i_1},…,a_{i_k}> \subset <a_1,a_2,…> $$
On the other hand for any word from $<a_1,a_2,…> $ I can construct a proper loop in $G$.
Is this a right way to do it?
Additionaly I'd like to ask how can I prove that $G$ is homotopy equivalent to infinite wedge sum of circles?
Can fundamental group of infinite wedge sum of circles be computed in other way then the above one?
Thank you for all your answers.
Best Answer
Your idea is right. Every element of $\pi_1(G)$ may be written as a word $r_{i_1}^{n_1} r_{i_2}^{n_2} \cdots r_{i_k}^{n_k}$ where $r_1, \cdots, r_n$ are the generators of $\pi_1(G)$ corresponding to the loop going around the $i$-th circle exactly once, for $i = 1, 2, \cdots, n$ (Notice that there may be multiple appearances of a single $r_k$ in the word, eg: $r_1r_2r_1^{-1}r_2^{-1}$ is a valid word; the point is $\pi_1(G)$ is not commutative) This is a well-defined assignment, and the word has finite length since as you said the image of a loop has to be compact. This in turn gives you a homomorphism $\pi_1(G) \to F(r_1, r_2, \cdots, r_n)$ which you can check is indeed an isomorphism.
As for why $G$ is homotopically equivalent to $\bigvee_{\Bbb N} S^1$, Consider the map $G \to \bigvee_{\Bbb N} S^1$ which quotients a neighborhood $U$ of the bad point. This is continuous, as can be easily checked. I think $U$ has a mapping cylinder neighborhood in $G$, which would imply $(G, U)$ has homotopy extension property. As $U$ is contractible, this means $G \to G/U$ is a homotopy equivalence, as desired (an explicit homotopy inverse should be $\bigvee_\Bbb N S^1 \to G$ given by sending circles to circles).