1. ## Fundamental Group Question

What is $\pi_1(S^1 \vee S^1 \vee S^2)$?

Is it $\mathbb{Z} * \mathbb{Z} * \{1\}$ by Van Kampen's Theorem?

Thanks, I just wanted to check this.

2. Yes, that is correct. It is a free abelian group on two generators.

3. Originally Posted by Erdos32212
What is $\pi_1(S^1 \vee S^1 \vee S^2)$?

Is it $\mathbb{Z} * \mathbb{Z} * \{1\}$ by Van Kampen's Theorem?

Thanks, I just wanted to check this.
Yes.

Since $S^2$ is simply connected, $\pi_1(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z} * \mathbb{Z}$.

The free product of a free group $\mathbb{Z}$ and a free group $\mathbb{Z}$ is a free group having a presentation $\{\alpha, \beta\}$, where $\alpha$ = 1 or -1 in the first $\mathbb{Z}$ and $\beta$ = 1 or -1 in the second $\mathbb{Z}$.

For instance,
$1_1 1_1 1_1 1_2 1_2$ (followed by a member in the first group) .... ,
reduces to
$3_1 2_2 ...$ ,

where $x_i$ denotes the element x in the i-th $\mathbb{Z}$, i=1 or 2, and the group operation between elements in the same group is an addition.

As you know, $\mathbb{Z} * \mathbb{Z}$ and $\mathbb{Z} \times \mathbb{Z}$ are two different things. The former is non-abelian because a free group on a nonempty set S is abelian iff S has exactly one element. The latter group is abelian, which is isomorphic to the fundamental group of a torus having a presentation $\{\alpha, \beta\ | \alpha\beta\alpha^{-1}\beta^{-1}=1\}$.