What is ?

Is it by Van Kampen's Theorem?

Thanks, I just wanted to check this.

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- May 2nd 2009, 10:50 AMErdos32212Fundamental Group Question
What is ?

Is it by Van Kampen's Theorem?

Thanks, I just wanted to check this. - May 2nd 2009, 02:30 PMGaloisTheory1
Yes, that is correct. It is a free abelian group on two generators.

- May 3rd 2009, 05:29 AMaliceinwonderland
Yes.

Since is simply connected, .

The free product of a free group and a free group is a free group having a presentation , where = 1 or -1 in the first and = 1 or -1 in the second .

For instance,

(followed by a member in the first group) .... ,

reduces to

,

where denotes the element x in the i-th , i=1 or 2, and the group operation between elements in the same group is an addition.

As you know, and 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 .