Fundamental Group Question

**Erdos32212** · May 2nd 2009, 09:50 AM

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.

**GaloisTheory1** · May 2nd 2009, 01:30 PM

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

**aliceinwonderland** · May 3rd 2009, 04:29 AM

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 $\{\alpha, \beta\ | \alpha\beta\alpha^{-1}\beta^{-1}=1\}$ .

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