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Math Help - Calculating Fundamental Groups

  1. #1
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    Calculating Fundamental Groups

    Calculate the fundamental groups of the following spaces:

    i) X_{1} = \{ (x, y, z) \in \mathbb{R}^3 | x>0 \}



    ii) X_{2} = \{ (x, y, z) \in \mathbb{R}^3 | x \neq 0 \}



    iii) X_{3} = \{ (x, y, z) \in \mathbb{R}^3 | (x, y, z) \neq (0, 0, 0) \}



    iv) X_{4} =  \mathbb{R}^3 \  \{ (x, y, z) | x=0, y=0, 0\leq z\leq1 \}



    v) X_{5} = \mathbb{R}^3\  \{ (x, y, z) | x=0, 0\leq y\leq 1 \}

    For i) \pi_{1}(X_{1}) = \{1\} as X_{1} is obviously simply connected so has trivial fundamental group.

    ii) As (X_{2}) is the the disjoint union of two contractible spaces I think this makes \pi_{1}(X_{2}) = \{1\} too?

    iii) and iv) are both homotopically equivalent to \mathbb{S}^2 and therefore simply connected so does that make them \{1\} also?

    for v) I have no idea?

    I'm just doubting myself as I find that they are just the trivial fundamental groups for the first four and would like some reassurance?

    any help on v) much appreciated too.
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  2. #2
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    Re: Calculating Fundamental Groups

    You're right for 1-4. For #5, it is equivalent to a cylinder. So the fundamental group is Z.
    Thanks from shmounal
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