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Math Help - Fundamental Group Question

  1. #1
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    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.
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  2. #2
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    Yes, that is correct. It is a free abelian group on two generators.
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  3. #3
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    Quote Originally Posted by Erdos32212 View Post
    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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