1. ## Hausdorff space, structure

Let $X$ be a Hausdorff space such that $X= A \cup B$ where $A$ and $B$ are each homeomorphic to a torus, and $A \cap B =\{x_0\}$.

What is the structure of $\pi_1(X, x_0)$?

How do I describe the structure of this topological space's fundamental group? I know how to draw it, etc. But, I don't know much about this space's fundamental group.

2. Originally Posted by Erdos32212
Let $X$ be a Hausdorff space such that $X= A \cup B$ where $A$ and $B$ are each homeomorphic to a torus, and $A \cap B =\{x_0\}$.

What is the structure of $\pi_1(X, x_0)$?

How do I describe the structure of this topological space's fundamental group? I know how to draw it, etc. But, I don't know much about this space's fundamental group.
Lemma 1. Let $X = U \cup V$, where U and V are open sets of X. If $U \cap V$ is simply connected, then $\pi(X)$ is the free product of groups $\pi(U)$ and $\pi(V)$ with respect to the homomorphisms $\phi_1 : \pi(U) \rightarrow \pi(X)$ and $\phi_2 : \pi(V) \rightarrow \pi(X)$.

If A and B were open subsets of X, we could apply Lemma 1 with U=A and V=B to determine the structure of $\pi_1(X, x_0)$.

Alternatively, we need to choose two open sets U and V such that A and B are deformation retracts of U and V, respectively.

Choose the circles and lines $S_1, l_1 \in A$ and $S_2, l_2 \in B$ such that if you remove the $S_1, l_1$ from A and $S_2,l_2$ from B, A and B should become simply connected (Think of a reverse procedure of making a torus from a rectangle paper).

Let $U=X - S_2 -l_2$ and $V=X - S_1 -l_1$, where $x_0$ does not belong to any $S_1, S_2, l_1$ and $l_2$. Then, A and B are deformation retracts of U and V, respectively

Now, we have $X = U \cup V$, and $U \cap V = X -S_1 -S_2 -l_1 -l_2$, where $U \cap V$ is contractible.

Then, by lemma 1, we conclude that $\pi_1(X, x_0)$ is the free product of groups $\pi_1(U, x_0)$ and $\pi_1(V, x_0)$, or the free product of groups $\pi_1(A, x_0)$ and $\pi_1(B, x_0)$.

Since $\pi_1(A) \cong Z \times Z$ and $\pi_1(B) \cong Z \times Z$, $\pi_1(X, x_0)$ is isomorphic to the free product of groups $Z \times Z$ and $Z \times Z$.